Random Walks in a One-Dimensional Lévy Random Environment

Bianchi, Alessandra; Cristadoro, Giampaolo; Lenci, Marco; Ligabò, Marilena

doi:10.1007/s10955-016-1469-0

Cited by 20 publications

(37 citation statements)

References 28 publications

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“…(27)(28)(29) has been tested in numerical simulations, as shown in Fig. 6, and then rigorously proved in a series of recent papers [49][50][51][52].…”

Section: A the Lévy Lorentz Gasmentioning

confidence: 99%

Single-big-jump principle in physical modeling

2019

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The big jump principle is a well established mathematical result for sums of independent and identically distributed random variables extracted from a fat tailed distribution. It states that the tail of the distribution of the sum is the same as the distribution of the largest summand. In practice, it means that when in a stochastic process the relevant quantity is a sum of variables, the mechanism leading to rare events is peculiar: instead of being caused by a set of many small deviations all in the same direction, one jump, the biggest of the lot, provides the main contribution to the rare large fluctuation. We reformulate and elevate the big jump principle beyond its current status to allow it to deal with correlations, finite cutoffs, continuous paths, memory and quenched disorder. Doing so we are able to predict rare events using the extended big jump principle in Lévy walks, in a model of laser cooling, in a scattering process on a heterogeneous structure and in a class of Lévy walks with memory. We argue that the generalized big jump principle can serve as an excellent guideline for reliable estimates of risk and probabilities of rare events in many complex processes featuring heavy tailed distributions, ranging from contamination spreading to active transport in the cell. PACS numbers:

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“…(27)(28)(29) has been tested in numerical simulations, as shown in Fig. 6, and then rigorously proved in a series of recent papers [49][50][51][52].…”

Section: A the Lévy Lorentz Gasmentioning

confidence: 99%

Single-big-jump principle in physical modeling

2019

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“…29 the dynamics of a kicked quantum system undergoing repeated measurements of momentum has been investigated. A diffusive behavior has been obtained, even when the dynamics of the classical counterpart is not chaotic, and in general, the system has been shown to have an anomalous diffusive behavior, characteristic of intermittent classical dynamical systems and random walks in random environments 30 .…”

Section: Introductionmentioning

confidence: 99%

Large-time limit of the quantum Zeno effect

Facchi

Ligabò

2017

Journal of Mathematical Physics

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If very frequent periodic measurements ascertain whether a quantum system is still in its initial state, its evolution is hindered. This peculiar phenomenon is called quantum Zeno effect. We investigate the large-time limit of the survival probability as the total observation time scales as a power of the measurement frequency, t ∝ N α .The limit survival probability exhibits a sudden jump from 1 to 0 at α = 1/2, the threshold between the quantum Zeno effect and a diffusive behavior. Moreover, we show that for α ≥ 1 the limit probability becomes sensitive to the spectral properties of the initial state and to arithmetic properties of the measurement periods.

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“…The expression 'Lévy random medium' indicates a stochastic point process, in some space, where the distances between nearby points have heavy-tailed distributions. Processes of this kind have been receiving a surge of attention, of late, both in the physical and mathematical literature; cf., respectively, [2,21,7,8,26,23] and [4,5,17,27]. They model a variety of situations that are of interest in the sciences.…”

Section: Introductionmentioning

confidence: 99%

Limit theorems for Lévy flights on a 1D Lévy random medium

Stivanello

Bet

Bianchi

et al. 2021

Electron. J. Probab.

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We study a random walk on a point process given by an ordered array of points (ω k , k ∈ Z) on the real line. The distances ω k+1 − ω k are i.i.d. random variables in the domain of attraction of a β-stable law, with β ∈ (0, 1) ∪ (1, 2). The random walk has i.i.d. jumps such that the transition probabilities between ω k and ω depend on − k and are given by the distribution of a Z-valued random variable in the domain of attraction of an α-stable law, with α ∈ (0, 1) ∪ (1, 2). Since the defining variables, for both the random walk and the point process, are heavy-tailed, we speak of a Lévy flight on a Lévy random medium. For all combinations of the parameters α and β, we prove the annealed functional limit theorem for the suitably rescaled process, relative to the optimal Skorokhod topology in each case. When the limit process is not càdlàg, we prove convergence of the finite-dimensional distributions. When the limit process is deterministic, we also prove a limit theorem for the fluctuations, again relative to the optimal Skorokhod topology.

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Random Walks in a One-Dimensional Lévy Random Environment

Cited by 20 publications

References 28 publications

Single-big-jump principle in physical modeling

Single-big-jump principle in physical modeling

Large-time limit of the quantum Zeno effect

Limit theorems for Lévy flights on a 1D Lévy random medium

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