Multidimensional Lévy walk and its scaling limits

Teuerle, Marek; Żebrowski, Piotr; Magdziarz, Marcin

doi:10.1088/1751-8113/45/38/385002

Cited by 31 publications

(30 citation statements)

References 51 publications

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“…Such annealed CTRWs give rise to the celebrated space-time fractional Fokker-Planck equations [5,19]. Another important class of CTRW models satisfying the renewal assumption, with additional property of coupling (strong dependence) between jumps and rests, are Lévy walks [20][21][22][23][24][25]. Applications of Lévy walks in the modeling of real-life phenomena include: fluid flow in a rotating annulus [6], blinking nanocrystals [26], Lévy-Lorentz gas [27], human travel [28,29], epidemic spreading [30,31], foraging of animals [32,33], transport of light in optical materials [34].…”

Section: Introductionmentioning

confidence: 99%

Quenched trap model for Lévy flights

Magdziarz

Szczotka

2016

Communications in Nonlinear Science and Numerical Simulation

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Section: Introductionmentioning

confidence: 99%

Quenched trap model for Lévy flights

Magdziarz

Szczotka

2016

Communications in Nonlinear Science and Numerical Simulation

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“…Then we have the following convergence of Lévy walks L(t), L U LW (t) and L OLW in Skorokhod space (which also implies convergence of all finite-dimensional distriubtions). I. Undershooting Lévy walk limit [31], [32]:…”

Section: Limit Processesmentioning

confidence: 99%

Method of calculating densities for isotropic ballistic Lévy walks

Magdziarz

Zorawik

2017

Communications in Nonlinear Science and Numerical Simulation

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We provide explicit formulas for asymptotic densities of d-dimensional isotropic Lévy walks, when d > 1. The densities of multidimensional undershooting and overshooting Lévy walks are presented as well. Interestingly, when the number of dimensions is odd the densities of all these Lévy walks are given by elementary functions. When d is even, we can express the densities as fractional derivatives of hypergeometric functions, which makes an efficient numerical evaluation possible.

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“…We can thus conclude [76,77] that the distribution of ||∆|| also has the heavy tail 1/z 2 . In the continuous time limit this leads to a superdiffusive, multidimensional LW or LF [76,[78][79][80][81][82]. Note that the components of ∆ are not independent and the spectral measure [76,81] takes a nontrivial form, which will be the subject of future studies.…”

Section: Figmentioning

confidence: 99%

Emergence of Lévy Walks from Second-Order Stochastic Optimization

Kuśmierz

Toyoizumi

2017

Phys. Rev. Lett.

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In natural foraging, many organisms seem to perform two different types of motile search: directed search (taxis) and random search. The former is observed when the environment provides cues to guide motion towards a target. The latter involves no apparent memory or information processing and can be mathematically modeled by random walks. We show that both types of search can be generated by a common mechanism in which Lévy flights or Lévy walks emerge from a second-order gradient-based search with noisy observations. No explicit switching mechanism is required -instead, continuous transitions between the directed and random motions emerge depending on the Hessian matrix of the cost function. For a wide range of scenarios the Lévy tail index is α = 1, consistent with previous observations in foraging organisms. These results suggest that adopting a second-order optimization method can be a useful strategy to combine efficient features of directed and random search.Many organisms must actively search for resources in order to survive and produce offspring. Foraging theory examines the various search strategies implemented by organisms depending on their abilities and the environments in which they live. In directed search, greater involvement of sensory and information processing abilities enable more complicated strategies. In contrast, in the boundary case of a memoryless and senseless forager, the only option is to wander randomly in the environment (random search). Even in this case, however, different strategies exist, depending on the character of the random motion. A natural candidate model for the random strategy is Brownian motion that describes a wide range of natural phenomena, including the movement of inanimate particles under thermal noise. A prominent feature of Brownian motion is the linear growth of the variance of the position with time. However, empirical data indicate that for organisms the observed growth is often faster. Lévy walks (LWs) [1][2][3] [22][23][24][25][26][27][28][29][30][31][32] and alternative superdiffusive models [33]. These observations have led to the so-called Lévy flight optimal foraging hypothesis, which states that LFs (or LWs) represent evolutionary adaptations due to their distinct advantages over other random search strategies [22,34,35].Recently this view has been disputed because none of the mentioned organisms is senseless and all of them are able to perform some forms of directed search (taxis), for example T cells and isolated bacteria perform chemotaxis [36][37][38][39][40], whereas fruit flies perform phototaxis [41], geotaxis [42], and chemotaxis [43,44]. Indeed, a number of studies have shown that characteristics of LFs and LWs may emerge naturally on large scales from more realistic case specific models of movement [45], including simple deterministic and semideterministic walks in complex environments [35,[46][47][48][49][50], self-avoiding random walks [51][52][53], diffusion with a timevarying diffusion constant [54][55][56], and a multiplicative, selfac...

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Multidimensional Lévy walk and its scaling limits

Cited by 31 publications

References 51 publications

Quenched trap model for Lévy flights

Quenched trap model for Lévy flights

Method of calculating densities for isotropic ballistic Lévy walks

Emergence of Lévy Walks from Second-Order Stochastic Optimization

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