Fractional Equations for the Scaling Limits of Lévy Walks with Position-Dependent Jump Distributions

Vassili N. Kolokoltsov ()
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Vassili N. Kolokoltsov: Faculty of Computation Mathematics and Cybernetics, Moscow State University, 119991 Moscow, Russia

Mathematics, 2023, vol. 11, issue 11, 1-19

Abstract: Lévy walks represent important modeling tools for a 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 have been 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 generalizations are analyzed.

Keywords: Lévy walks; fractional equations of variable order; Caputo–Dzherbashian and Riemann–Liouville derivatives; material fractional derivatives; scaling limit; continuous time random walks (CTRW); subordinated Markov processes (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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