Method of calculating densities for isotropic ballistic Lévy walks

Magdziarz, Marcin; Zorawik, Tomasz

doi:10.1016/j.cnsns.2016.11.026

Cited by 11 publications

(8 citation statements)

References 45 publications

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“…It is also worth to mention here that for d-dimensional rotationally invariant LW when d is odd, the densities of the limit processes are also given by elementary functions [20,21].…”

Section: Probability Distributions Of the Limit Processesmentioning

confidence: 99%

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Limit properties of Lévy walks

Magdziarz

Zorawik

2020

J. Phys. A: Math. Theor.

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In this paper we study properties of the diffusion limits of three different models of Lévy walks (LW). Exact asymptotic behavior of their trajectories is found using LePage series representation. We also prove an existing conjecture about total variation of LW sample paths. Based on this conjecture we verify martingale properties of the limit processes for LW. We also calculate their probability density functions and apply this result to determine the potential density of the associated non-symmetric α-stable processes. The obtained theoretical results for continuous LW can be used to recognize and verify this type of processes from anomalous diffusion experimental data. In particular they can be used to estimate parameters from experimental data using maximum likelihood methods.

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“…It is also worth to mention here that for d-dimensional rotationally invariant LW when d is odd, the densities of the limit processes are also given by elementary functions [20,21].…”

Section: Probability Distributions Of the Limit Processesmentioning

confidence: 99%

“…Interestingly, when d is odd, φ 1 is an elementary function [21]. The right fractional deriva- x (t 2 + 3 x 2 ) 5/2 1 (−t,t) (x).…”

Section: D+1-dimensional Process (M α (T) U α (T))mentioning

confidence: 99%

Limit properties of Lévy walks

Magdziarz

Zorawik

2020

J. Phys. A: Math. Theor.

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“…In the uniform model [4], the direction of the next flight is determined by choosing, randomly and uniformly, a point on a unit circle (on the surface of the unit sphere S d in the d-dimensional case [2,[5][6][7]). The resulting process is spatially isotropic and this allows to reduce the set of spatial variables to a single one, r = |r|.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic densities of planar Lévy walks: a non-isotropic case

Bystrik,

Denisov

2021

Preprint

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Lévy walks are a particular type of continuous-time random walks which results in a superdiffusive spreading of an initially localized packet. The original one-dimensional model has a simple schematization that is based on starting a new unidirectional motion event either in the positive or in the negative direction. We consider two-dimensional generalization of Lévy walks in the form of the so-called XY-model. It describes a particle moving with a constant velocity along one of the four basic directions and randomly switching between them when starting a new motion event. We address the ballistic regime and derive solutions for the asymptotic density profiles. The solutions have a form of first-order integrals which can be evaluated numerically. For specific values of parameters we derive an exact expression. The analytic results are in perfect agreement with the results of finite-time numerical samplings.

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“…The most popular method to analyze the PDFs of Lévy walk and CTRW model are the integral transform (Laplace or Fourier) methods. In [11][12][13], explicit inverse Fourier transform of the PDF of Lévy walk with constant velocity is given. In this paper we also give the PDF for Lévy walk even with non-constant velocity in terms of the Hermite polynomials.…”

Section: Introductionmentioning

confidence: 99%

Lévy walk with parameter dependent velocity: Hermite polynomial approach and numerical simulation

Xu¹,

Deng²,

Sandev³

2020

J. Phys. A: Math. Theor.

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To analyze stochastic processes, one often uses integral transform (Fourier and Laplace) methods. However, for the time-space coupled cases, e.g. the Lévy walk, sometimes the integral transform method may fail. Here we provide a Hermite polynomial expansion approach, being complementary to the integral transform method, to the Lévy walk. Two approaches are compared for some already known results. We also consider the generalized Lévy walk with parameter dependent velocity. Namely, we consider the Lévy walk with velocity which depends on the walking length or on the duration of each step. Some interesting features of the generalized Lévy walk are observed, including the special shapes of the probability density function, the first passage time distributions, and various diffusive behaviors of the mean squared displacement.

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Method of calculating densities for isotropic ballistic Lévy walks

Cited by 11 publications

References 45 publications

Limit properties of Lévy walks

Limit properties of Lévy walks

Asymptotic densities of planar Lévy walks: a non-isotropic case

Lévy walk with parameter dependent velocity: Hermite polynomial approach and numerical simulation

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