arXiv subjects

Tomasz Zorawik

Publications and source records attributed to Tomasz Zorawik.

6 recordsLinked to original sources

arXiv2016

Method of calculating densities for isotropic Lévy Walks

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.

math.PR↗

arXiv2015

Explicit Densities of Multidimensional Lévy Walks

We provide explicit formulas for asymptotic densities of the 2- and 3-dimensional ballistic Lévy walks. It turns out that in the 3D case the densities are given by elementary functions. The densities of the 2D Lévy walks are expressed in terms of hypergeometric functions and the right-side Riemann-Liouville fractional derivative which allows to efficiently evaluate them numerically. The theoretical results agree with Monte-Carlo simulations. The obtained functions solve certain differential equations with the fractional material derivative.

cond-mat.stat-mech↗

arXiv2015

Comment on "Fokker-Planck equations for nonlinear dynamical systems driven by non-Gaussian Lévy processes" [J. Math. Phys. 53, 072701 (2012)]

In an article [J. Math. Phys. 53, 072701 (2012)] X. Sun and J. Duan presented Fokker-Planck equations for nonlinear stochastic differential equations with non-Gaussian Lévy processes. In this comment we show a serious drawback in the derivation of their main result. In the proof of Theorem 1 in the aforementioned paper, a false assumption that each infinitely differentiable function with compact support is equal to its Taylor series, is used. We prove that although the derivation is incorrect, the result remains valid only if we add certain additional assumptions.

math-ph↗

arXiv2015

Stochastic representation of fractional subdiffusion equation. The case of infinitely divisible waiting times, Levy noise and space-time-dependent coefficients

In this paper we analyze fractional Fokker-Planck equation describing subdiffusion in the general infinitely divisible (ID) setting. We show that in the case of space-time-dependent drift and diffusion and time-dependent jump coefficient, the corresponding stochastic process can be obtained by subordinating two-dimensional system of Langevin equations driven by appropriate Brownian and Levy noises. Our result solves the problem of stochastic representation of subdiffusive Fokker-Planck dynamics in full generality.

math.PR↗

arXiv2015

Densities of Lévy walks and the corresponding fractional equations

In this paper we derive explicit formulas for the densities of Levy walks. Our results cover both jump-first and wait-first scenarios. The obtained densities solve certain fractional differential equations involving fractional material derivative operators. In the particular case, when the stability index is rational, the densities can be represented as an integral of Meijer G function. This allows to efficiently evaluate them numerically. Our results show perfect agreement with the Monte Carlo simulations.

math.AP↗

arXiv2015

Fourier-Bessel heat kernel estimates

We provide sharp two-sided estimates of the Fourier-Bessel heat kernel and we give sharp two-sided estimates of the transition probability density for the Bessel process in (0,1) killed at 1 and killed or reflected at 0.

math.CA↗