Escape from bounded domains driven by multivariate<i>α</i>-stable noises

Szczepaniec, Krzysztof; Dybiec, Bartłomiej

doi:10.1088/1742-5468/2015/06/p06031

Cited by 13 publications

(14 citation statements)

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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%

“…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. In contrast, if the shift of eigenvalue distribution related to the curvature and damping factor is strong enough, the tails can be cut off completely, due to the bounded support of the Wigner semicircle distribution.…”

Section: Figmentioning

confidence: 99%

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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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Section: Figmentioning

confidence: 99%

Section: Figmentioning

confidence: 99%

Emergence of Lévy Walks from Second-Order Stochastic Optimization

Kuśmierz

Toyoizumi

2017

Phys. Rev. Lett.

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“…dimension d with L = R) [32] because the average distance to the absorbing boundary is larger for the hypercube than for the hypersphere. For the hypersphere the MFPT decreases like a power-law with increasing value of the dimension d both for Cartesian and spherical Lévy flights, see [32]. For the hypercube, the dependence is not of the power-law type and follow more complex pattern.…”

Section: Escape In 2dmentioning

confidence: 99%

Escape from hypercube driven by multi-variate α-stable noises: role of independence

Dybiec

Szczepaniec

2015

Eur. Phys. J. B

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Abstract. We explore properties of the escape kinetics from the d-dimensional hypercube driven by multivariate α-stable noises. Using methods of stochastic dynamics we show complex dependence of the mean first passage time for the escape from the hypercube as a function of the hypercube dimension d. Finally, we show how the escape process can be used to quantify independence of components of multi-variate α-stable noises.

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“…E τ B(0,r) = r 2 2d , and ρ(x) is uniform distribution on the boundary of B(0, r), and the trajectories of the process hit ∂ B(0, r) in finite time with probability 1, because of the continuity and isotropy of Brownian motion. When X t is the α-stable Lévy process [12,15,16,17,18,19,20,21,22,23,24,25,26,27,28],…”

Section: Introductionmentioning

confidence: 99%

First exit and Dirichlet problem for the nonisotropic tempered $α$-stable processes

Liu,

Deng

2019

Preprint

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Escape from bounded domains driven by multivariateα-stable noises

Cited by 13 publications

References 84 publications

Emergence of Lévy Walks from Second-Order Stochastic Optimization

Emergence of Lévy Walks from Second-Order Stochastic Optimization

Escape from hypercube driven by multi-variate α-stable noises: role of independence

First exit and Dirichlet problem for the nonisotropic tempered $α$-stable processes

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