# Asymptotic densities of planar L\'{e}vy walks: a non-isotropic case

@inproceedings{Bystrik2021AsymptoticDO, title={Asymptotic densities of planar L\'\{e\}vy walks: a non-isotropic case}, author={Yu. S. Bystrik and Sergey Denisov}, year={2021} }

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… Expand

#### References

SHOWING 1-10 OF 30 REFERENCES

Asymptotic densities of ballistic Lévy walks.

- Mathematics, Medicine
- Physical review. E, Statistical, nonlinear, and soft matter physics
- 2015

It is shown that different scenarios of performing a random-walk step, via making an instantaneous jump penalized by a proper waiting time or via moving with a constant speed, dramatically effect the corresponding propagators, despite the fact that the end points of the steps are identical. Expand

Limit theorems for Lévy walks in d dimensions: rare and bulk fluctuations

- Physics, Mathematics
- 2017

We consider super-diffusive Levy walks in dimensions when the duration of a single step, i.e. a ballistic motion performed by a walker, is governed by a power-law tailed distribution of infinite… Expand

Method of calculating densities for isotropic ballistic Lévy walks

- Mathematics, Computer Science
- Commun. Nonlinear Sci. Numer. Simul.
- 2017

The authors provide explicit formulas for asymptotic densities of d-dimensional (d > 1) isotropic Levy walks in a ballistic regime and can express the densities as fractional derivatives of hypergeometric functions, which makes an efficient numerical evaluation possible. Expand

Infinite horizon billiards: Transport at the border between Gauss and Lévy universality classes.

- Physics, Medicine
- Physical review. E
- 2019

This work considers transport in two billiard models, the infinite horizon Lorentz gas and the stadium channel, presenting analytical results for the spreading packet of particles, and demonstrates the slow convergence rate via summation of independent and identically distributed random variables on the border between Lévy and Gauss laws. Expand

Explicit densities of multidimensional ballistic Lévy walks.

- Mathematics, Medicine
- Physical review. E
- 2016

This paper derives explicit formulas for densities of the two- and three-dimensional ballistic Lévy walks, which are most important in applications, and finds that in the 3D case the densities are given by elementary functions. Expand

Towards deterministic equations for Lévy walks: the fractional material derivative.

- Mathematics, Medicine
- Physical review. E, Statistical, nonlinear, and soft matter physics
- 2003

A generalized dynamical formulation is derived for Lévy walks, in which the fractional equivalent of the material derivative occurs, which is expected to be useful for the dynamical formulations of LÉvy walks in an external force field or in phase space. Expand

Random walks with infinite spatial and temporal moments

- Mathematics
- 1982

The continuous-time random walk of Montroll and Weiss has been modified by Scher and Lax to include a coupled spatial-temporal memory. We treat novel cases for the random walk and the corresponding… Expand

Dispersion of particles in an infinite-horizon Lorentz gas.

- Physics, Medicine
- Physical review. E
- 2018

An analytical solution is obtained for the Lorentz gas' kinetics on physically relevant timescales, and it is found that the density in its far tails decays as a universal power law of exponent -3. Expand

Superdiffusive Dispersals Impart the Geometry of Underlying Random Walks.

- Physics, Medicine
- Physical review letters
- 2016

It is shown that, in contrast to standard random walks, the microscopic geometry of planar superdiffusive Lévy walks is imprinted in the asymptotic distribution of the walkers. Expand

Measuring logarithmic corrections to normal diffusion in infinite-horizon billiards.

- Mathematics, Medicine
- Physical review. E, Statistical, nonlinear, and soft matter physics
- 2014

Numerically, it is shown that the expected asymptotic behavior of a weak form of superdiffusion in a two-dimensional periodic billiard model is easily overwhelmed by the subleading linear growth throughout the time range accessible to numerical simulations. Expand