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Discrete- and continuous-time random walks in 1D L\'evy random medium

A L\'evy random medium, in a given space, is a random point process where the distances between points, a.k.a. targets, are long-tailed. Random walks visiting the targets of a L\'evy random medium have been used to model many (physical, ecological, social) phenomena that exhibit superdiffusion as the result of interactions between an agent and a sparse, complex environment. In this note we consider the simplest non-trivial L\'evy random medium, a sequence of points in the real line with i.i.d. long-tailed distances between consecutive targets. A popular example of a continuous-time random walk in this medium is the so-called L\'evy-Lorentz gas. We give an account of a number of recent theorems on generalizations and variations of such model, in discrete and continuous time.


Introduction
In this note we give an overview of recent rigorous results on random walks (RWs) in random medium on the real line. The random medium is given by a point process ω = (ω k , k ∈ Z) ⊂ R, where ω 0 = 0 and the distances ζ k := ω k − ω k−1 between consecutive points are positive i.i.d. random variables with a long tail. By this we mean that the variance of ζ k is infinite. The points ω k will be henceforth called targets. For reason that will be better clarified below we refer to ω as a Lévy random medium.
We consider two types of RWs on R related to ω. To define them we introduce the auxiliary process S = (S n , n ∈ N), a Z-valued RW with S 0 = 0 and independent increments. We postulate that S is independent of ω and call it the underlying random walk. The first process of interest is Y ≡ Y ω := (Y n , n ∈ N), where Y n := ω Sn . This is the discrete-time RW (DTRW) that "jumps" on the targets of ω as determined by S. For example, if S produces the realization (0, 2, −3, . . .), the walker Y starts at the origin, then jumps to the second target to the right of 0, then to the third target to the left of 0, etc. The second process of interest is X ≡ X ω := (X(t), t ≥ 0), the continuous-time RW (CTRW) defined as the unitspeed interpolation of Y . This means that the walker X visits all the points Y n , ordered by n, but "walking" with unit speed rather than jumping. For instance, for S as in the above example, X starts at the origin and moves with velocity 1 until it reaches Y 1 = ω 2 , then it instantaneously turns its velocity to −1 and moves until it reaches Y 2 = ω −3 , and so on.
In the case where the underlying RW S is simple and symmetric, the process X is generally referred to as the Lévy-Lorentz gas, after Barkai, Fleurov and Klafter introduced it in the physical literature in 2000 [1]. The Lévy-Lorentz gas has been used since as a simple model for a number of phenomena exhibiting superdiffusion, i.e., diffusion at a faster speed than square root of time. They include transport in porous media, disordered optical media (such as Lévy glasses [2]), nanowires, etc.; see [1,3,4,5] and references therein.
In the physical literature, DTRWs, respectively CTRWs, whose distributions of jumps, respectively inertial stretches, are long-tailed, are often called Lévy flights, respectively Lévy walks. Lévy flights and walks, in regular or random media, have been employed as models for anomalous diffusion in a wide range of situations, from the physical to the biological and social sciences [6,7]. A rigorous mathematical treatment of these systems has only been given for the simplest of them, mostly on regular media. In real-world applications, however, the anomalous behavior of a certain diffusing quantity is seldom due to a special law governing the diffusing agent per se, but rather to the interaction between the agent and an irregular medium (e.g., a photon in a Lévy glass, a signal in a small-world network, an animal foraging where food is scarce, etc.). Hence the interest in Lévy media, namely, media that induce superdiffusive behavior, such as the random point process ω defined earlier.
A fair amount of mathematical work on the processes X and Y has been done in recent years by different authors [8,9,10,11,12]. In particular, in a number of cases, limit theorems have been proved for suitable rescalings of either process. In some instances, the convergence of moments has been proved as well. The purpose of this note is to present these results in a concise, unified manner. For reasons of space and self-consistency, we will neglect interesting work by Artuso and collaborators on yet another type of RW related to the Lévy-Lorentz gas, a persistent RW in an averaged medium [13,14].
In Section 2 we present results on the DTRW Y and in Section 3 on the CTRW X. Understandably, the results depend on the assumptions on the random medium ω and the underlying RW S. Major differences occur depending on whether ζ k , the distance between two consecutive targets, has infinite variance but finite mean, or infinite mean (and thus infinite variance), so we consider these cases in different subsections. No proofs are given, but references are placed throughout.

General notation
Throughout the paper we denote by P the probability law that governs the whole system, both the random medium ω and the underlying RW S (that latter playing the role of the random dynamics, as it "drives" both X and Y , in a given ω). P is called the annealed law and we denote by E its expectation. For a fixed ω, the conditional probability P ω := P(·|ω) is called the quenched law relative to the realization ω of the medium. We denote by E ω its expectation. A limit theorem, such as the CLT or the Invariance Principle, etc., relative to P is referred to as an annealed limit theorem. One speaks instead of a quenched limit theorem if the result is proved w.r.t. P ω , for P-a.e. ω. 1 In what follows, for the most part, we present annealed and quenched results in different subsections.
We indicate with ξ n := S n −S n−1 (n ∈ Z + ) the i.i.d. increments of the underlying RW, whose drift is denoted ν := E[ξ n ], if it exists in R ∪ {±∞}. The mean distance between the targets of ω is denoted µ := E[ζ k ]. Since ζ k > 0, µ always exists in R ∪ {+∞}. Let us recall that all ξ n (n ∈ Z + ) and ζ k (k ∈ Z) are independent.

Discrete-time random walk
In this section we consider the asymptotic behavior of the DTRW Y ≡ Y ω , under a number of different assumptions on the distributions of ζ 1 and ξ 1 .

Finite mean distance between targets, quenched theorems
We start with the results of [8] on the quenched version of Y , which only require a very simple condition on the medium, µ = E[ζ 1 ] < ∞, that is, the mean distance
Obviously, a quenched distributional limit theorem with the same limit for a.e. quenched law implies the annealed version of the same theorem: The limit in the statement of Theorem 2.1 holds w.r.t. P as well.
Convergence is known for the quenched moments of Y n / √ n as well, at least of lower order. Let p : For all q ∈ R + , denote by the q-th absolute moment of the standard Gaussian N (0, 1) (here Γ is the usual Gamma function). It is not hard to show that, at least for all q < p, The following is a reformulation of Theorem 2 of [8].
Theorem 2.3 Under the above assumptions and notation, fix q ∈ (0, p). For a.a. ω, Observe that µ q V q/2 ξ m q is the q-th absolute moment of N (0, µ 2 V ξ ), so Theorem 2.3 is consistent with Theorem 2.1.

Finite mean distance between targets, annealed theorems
In the next two subsections we report the functional limit theorems of [11]. We refer the reader to [16] for background material on stable laws, Lévy processes, Skorokhod topologies, etc. All distributional convergences in these subsections are meant w.r.t. P, that is, we are considering annealed functional limit theorems. We assume that ζ 1 is in the normal basin of attraction of a α-stable distribution, with α ∈ (1, 2]. This means that µ = E[ζ 1 ] < ∞ and (whose skewness index 2 must then be 0). As for the underlying RW, we assume ξ 1 is in the normal basin of attraction of an β-stable distribution, with β ∈ (0, 1) ∪ (1, 2]. We must distinguish two cases, depending on whether ν = E[ξ 1 ] exists and differs from 0, or otherwise.
• If β ∈ (0, 1), or β ∈ (1, 2] and ν = 0, we assume that there exists a β-stable variable W such that, as n → ∞, (2.4) • If β ∈ (1, 2] and ν = 0, we assume that there exists a β-stable variable W (β) 1 To state the results of this section, we need two spaces of functions with jump discontinuities. In what follows, we denote by D + the space of càdlàg 3 functions R + −→ R and by D the space of functions R −→ R whose restriction to [0, +∞), respectively (−∞, 0], is càdlàg, respectively càglàd. 3) (these conditions uniquely determine the distribution of the processes), and set By construction, every realization Z (α) belongs to D, and so do the realizations 3) entails functional convergence of these processes: as n → ∞, , relative to the Skorokhod topology J 1 . From now on, we will write 'in (D, J 1 )' for short.
We now introduce continuous-argument processes for the dynamics.
The following result extends Theorem 2.3 of [11].

Remark 2.5
The statement of [11,Thm. 2.3] does not include the cases α = 2 and/or β = 2, because the authors were mostly interested in bona fide Lévy media and Lévy flights in them. The proof of the theorem, however, works verbatim if the assumptions are generalized to include ζ 1 and/or ξ 1 in the normal domain of attraction of a 2-stable distribution, i.e., a Gaussian. In this case, of course, Z ± , W  [9] (which is stated for the case where S is simple and symmetric), but not quite, since the hypothesis on ζ 1 there is that P(ζ 1 > x) ≈ x −p , for x → +∞, with p ≥ 1. This is weaker than asking that ζ 1 be in the normal domain of attraction of an α-stable distribution, with α ∈ (1, 2]. For example it includes the case P(ζ 1 > x) = Cx −2 , where ζ 1 is in the domain, but not normal domain of attraction of a Gaussian [17,Ex. 5.10]. Also, the assertion of Theorem 2.4 is of course much stronger than that of Corollary 2.2, but the hypothesis on the medium for the latter is much weaker: simply µ < ∞.
The case (b) of the above theorem is the case where Y has a drift. Understandably, the scaling rate of Y ⌊n·⌋ is n (one says that the process is ballistic) and the convergence is to a deterministic function. It is therefore natural to study the fluctuations around the deterministic limit.

Remark 2.8
The limits in (b) and (c) involve the unusual Skorokhod topology J 2 [16, §11.5]. But this is the strongest amongst the classical Skorokhod topologies relative to which such limits hold, cf. Remark 2.11 of [11]. However, see Remark A.2 in the same paper.

Infinite mean distance between targets, annealed theorems
In this subsection we assume that ζ 1 is in the normal basin of attraction of a α-stable distribution with α ∈ (0, 1). Since E[ζ 1 ] = ∞, this means that, for n → ∞, for some α-stable variable Z (α) 1 (whose skewness index is 1, since ζ i > 0). Out of Z (α) 1 , we construct continuous-argument processes (Z It is a basic fact that, as n → ∞,ω (n) d − − → Z (α) , in (D, J 1 ). As the underlying RW, we maintain the same assumptions and notation as in §2.2, recalling that the fundamental assumption is that ξ 1 is in the normal basin of attraction of an β-stable distribution, with β ∈ (0, 1) ∪ (1, 2]. Once again, ν denotes the expectation of ξ 1 , when defined.

Remark 2.10
The convergence of the finite-dimensional distributions in assertion (a) is certainly a weak form of convergence, but it is morally the best one can do, given that Z (α) •W (β) is not càdlàg with positive probability; see the comments after Theorem 2.2 of [10]. As for assertion (b), the considerations of Remark 2.8 apply here too.

Continuous-time random walk
In this section we deal with the CTRW X ≡ X ω , again under various assumptions, depending on the papers we report on.

Finite mean distance between targets, quenched theorems
Once more, we start by presenting a result by [8], namely the quenched CLT for X [8, Thm. 1]. The assumptions are the same as in §2.1 above: the mean distance µ between targets is finite and the underlying RW has symmetric, unimodal, finitevariance increments ξ n . Let us recall in particular the notation V ξ := E[ξ 2 1 ]. Theorem 3.1 Under the above assumptions, most notably µ < ∞, let M ξ := E[|ξ 1 |] denote the first absolute moment of the underlying RW. Then, as t → ∞ and w.r.t. P ω , for a.a. ω, Here, once again, N (0, · ) is a centerd Gaussian variable with the specified variance.
The annealed CLT follows immediately: The limit in the statement of Theorem 3.1 holds w.r.t. P as well.
A recent preprint of Zamparo [12] claims the convergence of all quenched moments of X(t)/ √ t, under the additional assumption that the underlying RW is simple and symmetric (implying that X is the bona fide Lévy-Lorentz gas). Recall the notation m q for the q-th absolute moment of the standard Gaussian, cf. (2.1). Theorem 3.3 Assume that µ < ∞ and S is a simple symmetric RW. Then, for a.a. ω, Remark 3.4 When S is simple and symmetic, V ξ = M ξ = 1, so Theorem 3.3 is consistent with Theorem 3.1, showing that X is completely diffusive in this case. Theorem 3.3 descends from another result of independent interest, concerning the large deviations of X, namely, events of the type {|X(t)| > at}, for a > 0. Since X(t) is centered and scales like √ t, the probability of such "ballistic events" is expected to be exceedingly small. In [12,Thm. 2.3] it is proved that this probability vanishes like a stretched exponential. We report such result here:

Finite mean distance between targets, annealed theorems
Apart from recalling Corollary 3.2, which establishes the annealed CLT for X under the assumptions of [8] (µ < ∞ and S has symmetric, unimodal, finite-variance increments, cf. §2.1), in this section we present the results of [12] on the moments and large deviations of the annealed version of X.
The assumptions for this part are stronger than for Theorems 3.3 and 3.5. Like before, S must be a simple symmetric RW, but now we also posit that the tail of the distribution of ζ 1 is regularly varying with index −p ≤ −1. This means that where ℓ is a slowly varying function at +∞, namely, for all c > 0, In order to describe the upcoming theorems in their full power, we need more notation. For 0 < r < 1, set (3. 3) It can be seen [12, §2.1] that 0 < f p (r) ≤ r −p and, as r → 0 + , Here and in the rest of the paper ∼ denotes exact asymptotic equivalence. This limit shows in particular that 1 0 r q−1 f p (r) dr converges for q = 2p − 1 and p > 1, or q > 2p − 1 and p ≥ 1.
A few words of comment: In the subject of anomalous diffusion, an important quantity to investigate is the scaling exponent of the moments, log t , (3.5) assuming this limit exists at least for a.e. q > 0. In many relevant models one observes that q → γ(q) is piecewise linear with two branches, a left one with slope 1/2 and a right one with slope 1. Researchers named this situation strong anomalous diffusion, 5 cf. [19,6]. Theorem 3.6 shows that this is precisely what happens for the annealed Lévy-Lorentz gas, under the above assumptions. The corner between the two branches occurs at the moment of order 2p − 1 ≥ 2, so the behavior of the second moment is still normal, at least in terms of the leading exponent. Even more interestingly, this picture is very different from that of the corresponding quenched Lévy-Lorentz gas, which is fully diffusive, as seen in Theorem 3.3. 6 As for the quenched case, Theorem 3.6 is based on a large deviation result, which, however, is very different from Theorem 3.5: Theorem 3.7 Under the same assumptions as in Theorem 3.6, for a ∈ (0, 1], let Then, as t → ∞, P(X(t) > at) = P(X(t) < −at) ∼ F µ,p,a √ t τ ζ (t).
Remark 3.8 The first equality of the above assertion is obvious because, by the symmetry of the distributions of ω and S, the annealed distribution of X(t) is the same as that of −X(t).

Infinite mean distance between targets, annealed theorems
The last and hardest case is that of the CTRW X in a medium ω with µ = ∞. At least to this author's knowledge, only a limit theorem seems to be available, that of Bianchi et al, recently appeared in [10]. We present it after some preparatory material.
First off, the assumption on the medium is that ζ 1 is in the normal domain of attraction of an α-stable positive variable, with α ∈ (0, 1). As far ω is concerned, this is the same assumption as in §2.3, so we use the same notation introduced there, in particular for the processes (Z (α) ± (s), s ≥ 0) and (Z (α) (s), s ∈ R). The underlying random walk S is assumed to be centered and such that E[|ξ 1 | q ] < ∞, for some q > 2/α. This implies in particular that V ξ = E[|ξ 1 | 2 ] < ∞. So this is a special case of the assumptions on S of §2.3 (which were the same as in §2.2).
All the preliminary results seen earlier then apply, in particular, for n → ∞, (2) , in (D + , J 1 ). Here W (2) is a Brownian motion such that W (2) (t) has mean 0 and variance V ξ t. As clarified in §2.3, the processes Z (α) ± and W (2) are independent. Recalling the notation M ξ := E[|ξ 1 |], let ∆ := (∆(t), t ≥ 0) be defined by In other words, L t (x) is the local time of the Brownian motion W (2) in x, up to time t. As a function of x, L t is compactly supported and almost surely continuous, thus the above r.h.s. is welldefined. Since L t is also strictly increasing in t, ∆ is almost surely continuous and strictly increasing. Processes like ∆ are called Kesten-Spitzer processes and arise in the context of RW in random scenery [20], which is one of the technical ingredients of Theorem 2.1 of [10] which we now present.
Remark 3.10 The process Z (α) • W (2) • ∆ −1 is not a.s. càdlàg, so the same considerations and reference as in Remark 2.10 apply here.

A brief discussion on perspectives
Just by looking at the titles of the previous subsections, one notices that no quenched theorems were given for the case of infinite mean distance between targets. This is the main shortcoming of the current mathematical description of the processes X and Y . Technically speaking, the problem is that, without the condition µ < ∞, one does not have a strong law of large numbers for the variables ζ k . This is after all the simplest form of a quenched result and provides the scaling of k → ω k , as |k| → ∞, for each realization ω of the medium, apart from a negligible set of exceptions. How to prove quenched limit theorems without this basic ingredient is not clear to me at the moment.
An open question of a different nature is that of devising a good model of Lévy-Lorentz gas in dimension d ≥ 2. Here 'good' means that it should have the following features, in one form or another: • The random medium should be homogeneous, in the sense that the distribution of the relative positions of two or more targets should not depend on the absolute position of any of the targets involved. 7 • The distances between targets should be heavy-tailed. 8 • The law of the random medium should be rotation-invariant, at least for a subgroup of rotations, e.g., the coordinate directions. In other words, the model should be isotropic, unless it has a clear reason not to be.
• The transition probabilities from one target to the next should not depend on their distance, only on the "degree of accessibility" of the new target. For example, the next target might always be the nearest one along a random (isotropically selected) direction. The meaning of this condition is that it should be the medium, not the walker, to decide how long the next inertial stretch will be.
Even with all these features, a model might not be very interesting. Here is an example of a feasible, yet not very instructive model. Let (ω ′ k 1 , k 1 ∈ Z) and (ω ′′ k 2 , k 2 ∈ Z) be two i.i.d. point processes in R, as introduced in Section 1. For k = (k 1 , k 2 ) ∈ Z 2 , set ω k := (ω ′ k 1 , ω ′′ k 2 ). This defines a random medium ω = (ω k , k ∈ Z 2 ) in R 2 . An independent, Z 2 -valued, underlying RW (S n , n ∈ N) is given, whereby we introduce the DTRW Y := (Y n := ω Sn , n ∈ N). The CTRW X := (X(t), t ≥ 0) is then defined as the unit-speed interpolation of Y .
Obviously, the process X is simply the direct sum, in a very natural sense, of two independent, orthogonal, 1D CTRWs X ′ and X ′′ . Its properties are thus (for the most part) easily derived from those of X ′ and X ′′ , as presented in Section 3.
Introducing and investigating more relevant, and truly d-dimensional, flights and walks in Lévy random medium will be the subject of future work.