First passage and first hitting times of Lévy flights and Lévy walks

Palyulin, Vladimir V.; Blackburn, George L.; Lomholt, Michael A.; Watkins, Nicholas W.; Metzler, Ralf; Klages, Rainer; Chechkin, Aleksei V.

doi:10.1088/1367-2630/ab41bb

Cited by 74 publications

(54 citation statements)

References 137 publications

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“…In our analysis, we derive upper and lower bounds on the hitting time of a Lévy walk on Z 2 . Bounds on the hitting time and related quantities for Lévy walks on the (one-dimensional) real line are given in [29]. Bounds for general random walks on Z , for ≥ 1, in the case where the walk has bounded second (or higher) moments can be found in [37].…”

Section: Related Workmentioning

confidence: 99%

Search via Parallel Lévy Walks on Z2

Clementi

d'Amore

Giakkoupis

et al. 2021

Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing

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Motivated by the Lévy foraging hypothesis -the premise that various animal species have adapted to follow Lévy walks to optimize their search efficiency -we study the parallel hitting time of Lévy walks on the infinite two-dimensional grid. We consider independent discrete-time Lévy walks, with the same exponent ∈ (1, ∞), that start from the same node, and analyze the number of steps until the first walk visits a given target at distance ℓ. We show that for any choice of and ℓ from a large range, there is a unique optimal exponent ,ℓ ∈ (2, 3), for which the hitting time is˜(ℓ 2 / ) w.h.p., while modifying the exponent by any constant term > 0 increases the hitting time by a factor polynomial in ℓ, or the walks fail to hit the target almost surely. Based on that, we propose a surprisingly simple and effective parallel search strategy, for the setting where and ℓ are unknown: The exponent of each Lévy walk is just chosen independently and uniformly at random from the interval (2, 3). This strategy achieves optimal search time (modulo polylogarithmic factors) among all possible algorithms (even centralized ones that know ). Our results should be contrasted with a line of previous work showing that the exponent = 2 is optimal for various search problems. In our setting of parallel walks, we show that the optimal exponent depends on and ℓ, and that randomizing the choice of the exponents works simultaneously for all and ℓ. CCS CONCEPTS• Theory of computation → Distributed algorithms; • Mathematics of computing → Probabilistic algorithms.

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Section: Related Workmentioning

confidence: 99%

Search via Parallel Lévy Walks on Z2

Clementi

d'Amore

Giakkoupis

et al. 2021

Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing

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“…As can be seen, only for the case of Brownian motion (α = 2) the PDF has value zero at t = 0, while for LFs with α < 2 the first-passage time PDF exhibits a non-zero value at t = 0, thus demonstrating that LFs can instantly cross the boundary with their first jump away from their initial position x 0 . The magnitude of ℘(t → 0) can be estimated from the survival probability, as shown by equations ( 3) and (A.5) in [123] for symmetric LFs and here by equation ( 71) in section 5.2.5 below for asymmetric LFs with α ∈ (0, 2] and β ∈ (−1, 1] (excluding α = 1 with β = 0). Of course, in the case of symmetric LFs (β = 0) equation ( 71) coincides with equation (3) in [123].…”

Section: Symmetric Lfs In a Semi-infinite Domainmentioning

confidence: 99%

First passage time moments of asymmetric Lévy flights

Padash

Chechkin

Dybiec

et al. 2020

J. Phys. A: Math. Theor.

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We investigate the first-passage dynamics of symmetric and asymmetric Lévy flights in a semi-infinite and bounded intervals. By solving the space-fractional diffusion equation, we analyse the fractional-order moments of the first-passage time probability density function for different values of the index of stability and the skewness parameter. A comparison with results using the Langevin approach to Lévy flights is presented. For the semi-infinite domain, in certain special cases analytic results are derived explicitly, and in bounded intervals a general analytical expression for the mean first-passage time of Lévy flights with arbitrary skewness is presented. These results are complemented with extensive numerical analyses.

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“…Our analysis follows closely the one reported in del-Castillo-Negrete (1998) for the transport of passive scalars in vortices in the presence of a shear flow. Also, the study of exit time statistics of EP is closely related to the fundamental first-passage problem in non-equilibrium statistical mechanics, which is an open research problem in systems with nontrivial dynamics (see for example Benkadda et al (1997), Dybiec et al (2017), Palyulin et al (2019) and references therein). We follow during ∼10 7 cyclotron periods an ensemble of ∼10 5 energetic particles initialised with E ≈ 20E th in the chaotic region.…”

Section: Anomalous Exit Time and Asymmetric Diffusionmentioning

confidence: 99%

Anomalous losses of energetic particles in the presence of an oscillating radial electric field in fusion plasmas

Zarzoso

Del-Castillo-Negrete

2020

J. Plasma Phys.

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The confinement of energetic particles in nuclear fusion devices is studied in the presence of an oscillating radial electric field and an axisymmetric magnetic equilibrium. It is shown that, despite the poloidal and toroidal symmetries, initially integrable orbits turn into chaotic regions that can potentially intercept the wall of the tokamak, leading to particle losses. It is observed that the losses exhibit algebraic time decay different from the expected exponential decay characteristic of radial diffusive transport. A dynamical explanation of this behaviour is presented, within the continuous time random walk theory. The central point of the analysis is based on the fact that, contrary to the radial displacement, the poloidal angle is not bounded and a proper statistical analysis can therefore be made, showing for the first time that energetic particle transport can be super-diffusive in the poloidal direction and characterised by asymmetric poloidal displacement. The connection between poloidal and radial positions ensured by the conservation of the toroidal canonical momentum, implies that energetic particles spend statistically more time in the inner region of the tokamak than in the outer one, which explains the observed algebraic decay. This indicates that energetic particles might be efficiently slowed down by the thermal population before leaving the system. Also, the asymmetric transport reveals a new possible mechanism of self-generation of momentum.

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First passage and first hitting times of Lévy flights and Lévy walks

Cited by 74 publications

References 137 publications

Search via Parallel Lévy Walks on Z2

Search via Parallel Lévy Walks on Z2

First passage time moments of asymmetric Lévy flights

Anomalous losses of energetic particles in the presence of an oscillating radial electric field in fusion plasmas

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