Zero-one law for directional transience of one-dimensional random walks in dynamic random environments

We prove the trichotomy between transience to the right, transience to the left and recurrence of one-dimensional nearest-neighbour random walks in dynamic random environments under fairly general assumptions, namely: stationarity under space-time translations, ergodicity under spatial translations, and a mild ellipticity condition. In particular, the result applies to general uniformly elliptic models and also to a large class of non-uniformly elliptic cases that are i.i.d. in space and Markovian in time. An immediate consequence is the recurrence of models that are symmetric with respect to reflection through the origin.


Introduction
Random walks in random environments have been the subject of intensive mathematical study for several decades. They consist of random walks whose transition kernels are themselves random, modelling the movement of a tracer particle in a disordered medium. When the random transition kernels, called random environment, do not evolve with time, the model is called static; otherwise it is called dynamic. Hereafter we will use the abbreviations RWRE for the static and RWDRE for the dynamic model. The reader is referred to the monographs [19], [21] for RWRE and [2], [17] for RWDRE. Note that, by considering time as an additional dimension, one-dimensional RWDRE can be seen as directed RWRE in two dimensions.
In the present paper we consider the very basic question of whether the trichotomy between transience to the right, transience to the left and recurrence, typical for timehomogeneous Markov chains on Z, also holds for one-dimensional, nearest-neighbour RWDRE. We conclude that this is indeed the case under fairly general assumptions on the random environment. An immediate but interesting consequence is that reflectionsymmetric models satisfying our assumptions must be recurrent. We will consider the continuous time setting, but the same arguments work, mutatis mutandis, in the discrete time case. For comparison with other non-Markovian models where this problem was addressed, the reader is referred to [12] and [22] for the case of 2-dimensional RWRE, [16] for the case of random walks in Dirichlet environments, and to [1] for the case of 1-dimensional excited random walk.
The paper is organised as follows. In Section 2 we define our model and state our assumptions and results. Section 3 discusses our setup, providing examples and connections to the literature; the proof that a class of examples described therein fits our setting is postponed to Section 6. In Section 4, we present a graphical construction that will be useful in Section 5, where our main theorem is proved.

Model, assumptions and results
Let ω = (ω − t , ω + t ) t≥0 be a stochastic process taking values on ([0, ∞) Z ) 2 , called the dynamic random environment. We will assume that ω belongs to the space Ω of right-continuous paths from [0, ∞) to ([0, ∞) Z ) 2 , where the latter is endowed with the product topology. Given a realisation of ω, the RWDRE X = (X t ) t≥0 is defined as the time-inhomogeneous Markov jump process on Z whose laws (P ω x ) x∈Z satisfy The existence of such processes is standard (see e.g. [6], Chapter 4, Section 7). We give here a particular construction in Section 4 below. Without extra assumptions the process X may explode (i.e., make infinitely many jumps) in finite time; we thus enlarge the state-space Z with a cemetery point ∆ in the standard way in order to define X after the explosion time τ ∆ , i.e., X t := ∆ for all t ≥ τ ∆ (cf. (4.3)). The law P ω x is called the quenched law. We denote by P x the joint law of X and ω (with P x (X 0 = x) = 1). The corresponding expectations will be denoted respectively by E ω x and E x . In the literature, the annealed (or averaged ) law is often defined as the marginal law of X under P x , but for convenience we will call P x itself the annealed law.
Define the space-time translation operators θ z s : Ω → Ω, z ∈ Z, s ∈ R + , acting on ω as (θ z s ω) x,t := ω(z + x, s + t). We will write θ s := θ 0 s , θ z := θ z 0 . Denoting by the natural filtration of X, the Markov property for X then reads for any bounded measurable f and any t ≥ 0. Moreover, since the space-time process (X t , t) is Feller, by the strong Markov property the time t in (2.3) may be replaced by any a.s. finite F t -stopping time. Also, we may and will assume that X is right-continuous.
We will work under the following assumptions: The process ω is stationary with respect to space-time translations, i.e., for each z ∈ Z, t ≥ 0, θ z t ω has the same distribution as ω. Furthermore, we assume that ω is ergodic with respect to the spatial translations θ z .  Assumption (SE) is standard; in fact, ω is usually taken ergodic also in time. Assumption (EL) is an ellipticity condition; note that it holds e.g. when ω is uniformly elliptic, i.e., if there exists κ ∈ (0, 1) such that κ ≤ ω ± t (x) ≤ κ −1 . Indeed, in this case the property ECP 21 (2016), paper 15. of being visited infinitely often is either a.s. satisfied by all or by none of the points of Z.
Note that (EL) implies inf{t > 0 : X t ∈ [−n, n] c } < τ ∆ P 0 -a.s. for all n ∈ N. We can now state our main result.
Theorem 2.1. If assumptions (SE) and (EL) are satisfied, then τ ∆ = ∞ P 0 -a.s. and one of the following three cases holds: as the ellipticity assumption (EL) ensures that the event appearing in item 3 is almost surely equal to the complement of the union of the events in 1-2.
As an immediate consequence of Theorem 2.1, we obtain recurrence for any model satisfying (SE)-(EL) that is symmetric with respect to reflection through the origin: Assume that (SE) and (EL) hold and that (−X t ) t≥0 has under P 0 the same distribution as (X t ) t≥0 . Then item 3 of Theorem 2.1 holds.
For an interesting example to which Corollary 2.2 applies, consider the following. Let is a simple symmetric exclusion process in Z started from a product Bernoulli measure ν ρ with density ρ ∈ (0, 1). Very little is known for this model in the literature (see e.g. [3], [10] and [18]), in particular in the case ρ = 1/2 where the expected asymptotic speed of X is zero. However, since it satisfies (SE)-(EL) and is reflection-symmetric for ρ = 1/2, Corollary 2.2 implies that it is recurrent in this case.

Examples and discussion
In the literature, ω is often given as a functional of an interacting particle system, i.e., of a Markov process (η t ) t≥0 on E Z where E is a metric space, often assumed compact.
For example, in the setting of [15], the transition rates are given by where the functions α ± : E Z → [0, ∞) satisfy some regularity properties. The setting of [4] is a particular case where E = {0, 1}.
Since directional transience follows from a law of large numbers with non-zero speed, and recurrence from a functional central limit theorem if the speed is zero, Theorem 2.1 brings no new information in the cases where these results are known. However, our result applies to many situations where such theorems have not yet been proved, which is the case for several uniformly elliptic but non-uniformly mixing models, e.g., when η t is a simple exclusion process or a system of independent random walks outside the perturbative regimes considered in [7], [10]. By "uniformly mixing" here we mean that the conditional law of η t (0) given η 0 converges to a fixed law uniformly over all possible ECP 21 (2016), paper 15. realizations of η 0 ; cf. e.g. the cone-mixing condition of [4] (Definition 1.1 therein), or the coupling conditions of [15] (Assumptions 1a-1b therein).
Let us now describe a large class of examples satisfying our assumptions that includes many non-uniformly elliptic cases with slow and non-uniform mixing: The proof of this theorem is given in Section 6 below. Note that, as already mentioned, it covers many models that are slowly and non-uniformly mixing and thus do not fall into the categories generally studied in the literature of RWDRE so far.
It is interesting to ask in which directions Theorem 2.1 could be generalised, and how far our hypotheses could be weakened. The analogous result in discrete time can be proved with a similar approach via graphical representation (cf. Section 4 below).
However, new ideas are needed for random walks in other graphs, e.g. Z d with nonnearest neighbour jumps and/or d > 1, and regular trees.

Graphical construction
We construct next a particular version of the process X with convenient properties. (4.1) We denote by P ω the joint law of N + ω , N − ω , and by P the joint law of the latter and ω. Define the space-time translations θ z t of N ± ω and functions thereof by We note that, under P, N ± ω inherits from ω the stationarity with respect to space-time translations and the ergodicity with respect to spatial translations.
On each point of N + ω , resp. N − ω , we draw a unit-length arrow pointing to the right, resp. to the left. Then we set, for x ∈ Z, X x to be the path started at x that proceeds by moving upwards in time and forcibly across any arrows in a right-continuous way; the paths are defined only up to the explosion time. See Figure 1.
Using the right-continuity of ω, it is straightforward to check that this construction gives the correct law, i.e., X x has under P ω the same law as X under P ω x . In particular, this provides a coupling for copies of the random walk starting from all initial positions, which will facilitate the proof of Theorem 2.1.
With this construction, the explosion times τ x ∆ , x ∈ Z can be defined as τ x ∆ := sup{t > 0 : X x crosses finitely many arrows up to time t}, and we identify X := X 0 , τ ∆ := τ 0 ∆ . We end this section with the following monotonicity property, which is a consequence of the graphical construction and will be useful in the proof of Lemma 5.3 below.
Lemma 4.1. For any y, z ∈ Z such that y ≤ z, P-a.s., Proof. Since the paths start ordered, move by nearest-neighbour jumps, and a.s. cannot jump simultaneously before they meet, either X y t < X z t for all relevant t or there exists a first s ≥ 0 such that X y s = X z s , in which case by construction X y u = X z u for all u ≥ s.

Proof of Theorem 2.1
For A ⊂ Z, denote by H A := inf{t > 0 : X t ∈ A} z . Our proof of Theorem 2.1 is based on three lemmas which we state next; their proofs are given respectively in Sections 5.1 and 5.2 below. The first of them implies that, if the random walk visits −1 (resp. 1) a.s., then all its excursions from 0 to the right (resp. to the left) will be a.s. finite.

Lemma 5.1. Assume that (SE) holds, and let
s. for all k ≥ 1.  The third lemma shows that, if there is a positive probability for the random walk to never touch −1 (resp. 1), then its range is bounded from below (resp. above).
Note that Lemmas 5.1-5.3 do not use assumption (EL) directly but only its consequence (2.5). Moreover, Lemma 5.1 only uses stationarity in time and the strong Markov property; the graphical construction of Section 4 is only used in the proof of Lemmas 5.2 and 5.3.
We can now finish the proof of Theorem 2.1.
Proof of Theorem 2.1. Assumption (EL) and Lemmas 5.2-5.3 together imply that since, if the left-hand side of (5.6) holds, then lim inf t→∞ X t > −∞ a.s. and hence it must be equal to ∞ by (EL). Analogously, To conclude, we claim that Indeed, note that, by Lemmas 5.1-5.2, P 0 (H −1 < ∞) = 1 implies that lim inf t→∞ X t ≤ −1 a.s., which together with (EL) gives lim inf t→∞ X t = −∞. The last equality is obtained analogously.

Proof of Lemma 5.1
Proof. To start, we claim that, P 0 -a.s., P θtω 0 (H x = ∞) = 0 simultaneously for all t ≥ 0. (5.9) Indeed, for each fixed t ≥ 0, P θtω 0 (H x = ∞) = 0 a.s. since, by stationary, P 0 (·) = E 0 [P ω 0 (·)] = E 0 [P θtω 0 (·)]. Hence (5.9) holds with t restricted to the set of rational numbers, and to extend it to all t ≥ 0 we only need to show that the function t → P θtω 0 (H x = ∞) is right-continuous. To this end, note that, since ω is right-continuous, for all s > 0 small enough (depending on ω and t). Denoting by O ω (s) a function whose modulus is bounded by s C ω where C ω ∈ (0, ∞) may depend on ω, we obtain where for the second line we use the Markov property and for the last one we again use (5.10). From this follows the desired right-continuity and consequently also (5.9). By the strong Markov property (cf. the paragraph below (2.3)) and Θ T (k) by (5.9).

Proof of Lemmas 5.2-5.3
We start by showing that explosions are not possible under (SE) and (2.5).

Proof of Lemma 5.2.
It is enough to show that, for any a, b ∈ [0, ∞) with b − a > 0 small enough, To that end, define the events and let ε > 0 be so small that, if b − a ≤ ε, then (5.15) which exists by the right-continuity of ω and the dominated convergence theorem. Noting that A a,b x = θ x A a,b 0 , we obtain from Birkhoff's ergodic theorem that, P-a.s., by stationarity under time translations, and analogously for x ≤ 0. In particular, P ∀ z ∈ Z, ∃ x < z < y such that A a,b x and A a,b y occur = 1. (5.17) Note now that, by (2.5), if τ ∆ ∈ (a, b] then for all n ∈ N the random walk exits the interval [−n, n] before time b. Therefore where H [−n,n] c is the hitting time of Z\[−n, n] by X 0 . On the other hand, by the graphical construction, if both A a,b x and A a,b y occur for some x < X 0 a < y, then a + Θ a H [−n,n] c ≥ b with n = |x| ∨ |y|. Hence (5.18) is at most P ∀ x, y ∈ Z such that x < X 0 a < y, either A a,b x or A a,b y does not occur = 0 (5.19) by (5.17), proving (5.13).
We now prove Lemma 5.3.

Proof of Theorem 3.2
We first show that (2.5) holds for a very large class of models, including our examples.
Proof. Fix n ∈ N. By right-continuity and invariance under time translations, there exist δ, ε ∈ (0, 1) such that the events Note now that, if X V k ∈ [−n, n], then between times V k and V k + ε ∧ Θ V k H [−n,n] c the RWDRE has a rate at least δ to jump in direction . Therefore, for some deterministic ϑ n ∈ (0, 1) independent of k. By the Markov property, and we conclude using induction that (6.6) is equal to zero, proving (2.5). Proof. The models described are mixing in time, and thus satisfy the hypotheses of Proposition 6.1. Since they also satisfy (SE), the corollary follows by Lemma 5.2.
We can now finish the proof of Theorem 3.2.
Proof of Theorem 3.2. We will prove that, P 0 -a.s., The analogous result for x + 1 in place of x − 1 then follows by reflection. This implies (EL) since, by Proposition 6.1, (2.5) holds. We claim that it suffices to show that, a.s., Indeed, fix j ∈ N. Suppose by induction that (6.10) holds with (1) substituted by (j).