Kolokoltsov, V.N. Fractional Equations for the Scaling Limits of Lévy Walks with Position-Dependent Jump Distributions. Mathematics2023, 11, 2566.
Kolokoltsov, V.N. Fractional Equations for the Scaling Limits of Lévy Walks with Position-Dependent Jump Distributions. Mathematics 2023, 11, 2566.
Kolokoltsov, V.N. Fractional Equations for the Scaling Limits of Lévy Walks with Position-Dependent Jump Distributions. Mathematics2023, 11, 2566.
Kolokoltsov, V.N. Fractional Equations for the Scaling Limits of Lévy Walks with Position-Dependent Jump Distributions. Mathematics 2023, 11, 2566.
Abstract
L\'evy walks represent important modeling tools for variety of real life processes. Their natural scaling limits are known to be described by the so-called material fractional derivatives. So far these scaling limits were derived for spatially homogeneous walks, where Fourier and Laplace transforms represent natural tools of analysis. Here we derive the corresponding limiting equations in the case of position depending times and velocities of walks, where Fourier transforms cannot be effectively applied. In fact, we derive three different limits (specified by the way the process is stopped at an attempt to cross the boundary), leading to three different multi-dimensional versions of Caputo-Dzherbashian derivatives, which correspond to different boundary conditions for the generators of the related Feller semigroups and processes. Some other extensions and generalisations are analysed.
Keywords
L´evy walks; fractional equations of variable order; Caputo-Dzherbashian and Riemann-Liouville derivatives; material fractional derivatives; scaling limit; continu-ous time random walks (CTRW); subordinated Markov processes
Subject
Computer Science and Mathematics, Probability and Statistics
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
