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

Author

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  • Vassili N. Kolokoltsov

    (Faculty of Computation Mathematics and Cybernetics, Moscow State University, 119991 Moscow, Russia)

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.

Suggested Citation

  • Vassili N. Kolokoltsov, 2023. "Fractional Equations for the Scaling Limits of Lévy Walks with Position-Dependent Jump Distributions," Mathematics, MDPI, vol. 11(11), pages 1-19, June.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:11:p:2566-:d:1163245
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    References listed on IDEAS

    as
    1. Straka, P. & Henry, B.I., 2011. "Lagging and leading coupled continuous time random walks, renewal times and their joint limits," Stochastic Processes and their Applications, Elsevier, vol. 121(2), pages 324-336, February.
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