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

Dybiec, Bartłomiej; Szczepaniec, Krzysztof

doi:10.1140/epjb/e2015-60429-2

Cited by 6 publications

(5 citation statements)

References 38 publications

(49 reference statements)

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“…Extreme FPTs for diffusive search were first studied in 1983 by Weiss et al [48]. Driven primarily by applications to cell biology, extreme FPTs for diffusion have been recently studied by several groups of authors [21,44,[49][50][51][52][53][54][55][56][57]. Extreme FPTs for diffusion have also recently been generalized to a framework involving depletion of a stock of resources [58].…”

Section: Discussionmentioning

confidence: 99%

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Extreme hitting probabilities for diffusion*

Linn¹,

Lawley²

2022

J. Phys. A: Math. Theor.

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A variety of systems in physics, chemistry, biology, and psychology are modeled in terms of diffusing ``searchers'' looking for ``targets.'' Examples range from gene regulation, to cell sensing, to human decision-making. A commonly studied statistic in these models is the so-called hitting probability for each target, which is the probability that a given single searcher finds that particular target. However, the decisive event in many systems is not the arrival of a given single searcher to a target, but rather the arrival of the fastest searcher to a target out of many searchers. In this paper, we study the probability that the fastest diffusive searcher hits a given target in the many searcher limit, which we call the extreme hitting probability. We first prove an upper bound for the decay of the probability that the searcher finds a target other than the closest target. This upper bound applies in very general settings and depends only on the relative distances to the targets. Furthermore, we find the exact asymptotics of the extreme hitting probabilities in terms of the short-time distribution of when a single searcher hits a target. These results show that the fastest searcher always hits the closest target in the many searcher limit. While this fact is intuitive in light of recent results on the time it takes the fastest searcher to find a target, our results give rigorous, quantitative estimates for the extreme hitting probabilities. We illustrate our results in several examples and numerical solutions.

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Section: Discussionmentioning

confidence: 99%

“…The intuitive reasoning behind (49) is the same as (48). Namely, in the small target limit, even the fastest searcher wanders around the entire domain before finding the target.…”

Section: Narrow Capture In Three Dimensionsmentioning

confidence: 98%

Extreme hitting probabilities for diffusion*

Linn¹,

Lawley²

2022

J. Phys. A: Math. Theor.

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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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“…In particular, we will show that, thanks to various distributional and path properties of stable processes, notably spatial homogeneity, isotropy, self-similarity and that it exits D by a jump, simulations can be made unbiased, without the need to truncate the algorithm at an ε tolerance. Whilst there exist many methods for numerically examining the fractional Dirichlet problem (1.7), which mostly appeal to classical methodology for diffusive operators, see for example (Acosta, Borthagaray, Bruno, and Maas, 2016;D'Elia and Gunzburger, 2016;Dybiec and Szczepaniec, 2015;Huang and Oberman, 2014;Nochetto, Otárola, and Salgado, 2016;Szczepaniec and Dybiec, 2015;Zoia, Rosso, and Kardar, 2007) to name some but not all of the existing literature, we believe that no other work appeals to the walk-on-spheres algorithm in this context.…”

Section: Introductionmentioning

confidence: 99%

Unbiased ‘walk-on-spheres’ Monte Carlo methods for the fractional Laplacian

Kyprianou

Osojnik

Shardlow

2017

IMA Journal of Numerical Analysis

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We consider Monte Carlo methods for simulating solutions to the analogue of the Dirichlet boundary-value problem in which the Laplacian is replaced by the fractional Laplacian and boundary conditions are replaced by conditions on the exterior of the domain. Specifically, we consider the analogue of the so-called 'walk-on-spheres' algorithm. In the diffusive setting, this entails sampling the path of Brownian motion as it uniformly exits a sequence of spheres maximally inscribed in the domain. As this algorithm would otherwise never end, it is truncated when the 'walk-on-spheres' comes within ε > 0 of the boundary. In the setting of the fractional Laplacian, the role of Brownian motion is replaced by an isotropic α-stable process with α ∈ (0, 2). A significant difference to the Brownian setting is that the stable processes will exit spheres by a jump rather than hitting their boundary. This difference ensures that disconnected domains may be considered and that, unlike the diffusive setting, the algorithm ends after an almost surely finite number of steps.issued from x ∈ D, then, by the strong law of large numbers,For practical purposes, since it is impossible to take the limit, one truncates the series of estimates for large n and the central limit theorem gives O(1/n) upper bounds on the variance of the n-term sum, which serves as a numerical error estimate. Although forming the fundamental basis of most Monte Carlo methods for diffusive Dirichlettype problems, (1.3) is an inefficient numerical approach. Least of all, this is because the Monte Carlo simulation of u(x) is independent for each x ∈ D. Moreover, it is unclear how exactly to simulate the path of a Brownian motion on its first exit from D, that is to say, the quantity W τ D . This is because of the fractal properties of Brownian motion, making its path difficult to simulate. This introduces additional numerical errors over and above that of Monte Carlo simulation.A method proposed by (Muller, 1956), for the case that D is convex, sub-samples special points along the path of Brownian motion to the boundary of the domain D. The method does not require a complete simulation of its path and takes advantage of the distributional symmetry of Brownian motion. In order to describe the so-called 'walk-on-spheres', we need to first introduce some notation. We may thus set ρ 0 = x for x ∈ D and define r 1 to be the radius of the largest sphere inscribed in D that is centred at x. This sphere we will call S 1 = {y ∈ R d : |y − ρ 0 | = r 1 }. To avoid special cases, we henceforth assume that the surface area of S 1 ∩ ∂D is zero (this excludes, for example, the case that x = 0 and D is a sphere centred at the origin). Now set ρ 1 ∈ D to be a point uniformly distributed on S 1 and note that, given the assumption in the previous sentence, P x (ρ 1 ∈ ∂D) = 0. Construct the remainder of the sequence (ρ n , n ≥ 1) inductively. Given ρ n−1 , we define the radius, r n , of the largest sphere inscribed in D that is centred at ρ n−1 . Calling this sphere S n , we have that ...

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

Cited by 6 publications

References 38 publications

Extreme hitting probabilities for diffusion*

Extreme hitting probabilities for diffusion*

Emergence of Lévy Walks from Second-Order Stochastic Optimization

Unbiased ‘walk-on-spheres’ Monte Carlo methods for the fractional Laplacian

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