The Range of Stable Random Walks

We consider trap models in Z d . These are stochastic processes in a random environment as follows. The environment is given by a family τ = (τ x , x ∈ Z d ) of positive iid random variables in the basin of attraction of an α-stable law, 0 < α < 1. Given τ , our process is a continuous time Markov pure jump process, whose jump chain is a in principle generic random walk in Z d , d ≥ 1, independent of τ , and τ represents the holding time averages of the continuous time process. We may think of the sites of Z d as traps, and of τ x as the depth of trap x. We are interested in the trap process, namely the process that associates to time t the depth of the currently visited trap. Our first result is the convergence of the law of that process under suitable scaling. The limit process is given by the jumps of a certain α-stable subordinator at the inverse of another α-stable subordinator, correlated with the first subordinator. For that result, the requirements on the underlying random walk are a) the validity of a law of large numbers for its range, and b) the slow variation at infinity of the tail of the distribution of its time of return to the origin: they include all transient random walks as well as all random walks in d ≥ 2, and also many one dimensional random walks, but not the simple symmetric case. We then derive aging results for our process, namely scaling limits for some two-time correlation functions of the process; a strong form of those results requires an assumption of transience, stronger than a, b above. The scaling limit result mentioned above is an averaged result with respect to the environment. Under an additional condition on the size of the intersection of the ranges of two independent copies of the underlying random walk, roughly saying that it is small compared with size of the range, we derive a stronger scaling limit result, roughly stating that it holds in probability with respect to the environment. With that additional condition, we also strengthen the aging results, from the averaged version mentioned above, to convergence in probability with respect to the environment.

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“…But all planar random walks [21,22], and 1-dimensional β-stable random walks with β ≤ 1 [24] also satisfy Assumptions A and B.…”

Section: Remark 2 All Transient Random Walks Inmentioning

confidence: 99%

On the dynamics of trap models in ℤ^d

Fontes

Mathieu

2013

Proceedings of the London Mathematical Society

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“…The case α = 1 is a particular case since in that situation it is in general not known whether the random walk is transient or recurrent. The second and the third statement were proved by Le Gall and Rosen (1991). We mention that Le Gall and Rosen (1991) only state the marginal convergence in R3.…”

Section: Introductionmentioning

confidence: 88%

“…The second and the third statement were proved by Le Gall and Rosen (1991). We mention that Le Gall and Rosen (1991) only state the marginal convergence in R3. However, their proof also covers the joined convergence described in R3.…”

Section: Introductionmentioning

confidence: 88%

“…It is obvious that the distribution of K n depends on the number of sites that the random walk {S k , k ∈ N} has visited until time n. In fact it is the range R n := card{S 1 , ..., S n } of the underlying random walk {S k , k ∈ N} which determines the behaviour of the sequence {K n , n ∈ N}. The asymptotic behaviour of the range for rescaled integervalued random walks S (n) defined in (1) can be found in Le Gall and Rosen (1991). They present the following results, if S (n) converges toward an α-stable Lévy-process Y : R1.…”

Section: Introductionmentioning

confidence: 99%

“…α > 1 (see Spitzer (1976)). For 2/3 ≤ α < 1 one knows that h(n) → q −1 as n → ∞, and (R1) yields the almost sure convergence of h(n) n R [nt] toward t (see Le Gall and Rosen (1991)). The case α = 1 is a particular case since in that situation it is in general not known whether the random walk is transient or recurrent.…”

Section: Introductionmentioning

confidence: 99%

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The extremes of random walks in random sceneries

Franke

Saigo

2009

Adv. Appl. Probab.

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In this article we analyse the behaviour of the extremes of a random walk in a random scenery. The random walk is assumed to be in the domain of attraction of a stable law and the scenery is assumed to be in the domain of attraction of an extreme value distribution. The resulting random sequence is stationary and strongly dependent if the underlying random walk is recurrent. We prove a limit theorem for the extremes of the resulting stationary process. However, if the underlying random walk is recurrent, the limit-distribution is not in the class of classical extreme value distributions.

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