Article
Version 1
Preserved in Portico This version is not peer-reviewed
Biased Continuous-Time Random Walks with Mittag-Leffler Jumps
Version 1
: Received: 1 October 2020 / Approved: 6 October 2020 / Online: 6 October 2020 (10:34:24 CEST)
A peer-reviewed article of this Preprint also exists.
Michelitsch, T.M.; Polito, F.; Riascos, A.P. Biased Continuous-Time Random Walks with Mittag-Leffler Jumps. Fractal Fract. 2020, 4, 51. Michelitsch, T.M.; Polito, F.; Riascos, A.P. Biased Continuous-Time Random Walks with Mittag-Leffler Jumps. Fractal Fract. 2020, 4, 51.
Abstract
We construct admissible circulant Laplacian matrix functions as generators for strictly increasing random walks on the integer line. These Laplacian matrix functions refer to a certain class of Bernstein functions. The approach has connections with biased walks on digraphs. Within this framework, we introduce a space-time generalization of the Poisson process as a strictly increasing walk with discrete Mittag-Leffler jumps subordinated to a (continuous-time) fractional Poisson process. We call this process ‘space-time Mittag-Leffler process’. We derive explicit formulae for the state probabilities which solve a Cauchy problem with a Kolmogorov-Feller (forward) difference-differential equation of general fractional type. We analyze a “well-scaled” diffusion limit and obtain a Cauchy problem with a space-time convolution equation involving Mittag-Leffler densities. We deduce in this limit the ‘state density kernel’ solving this Cauchy problem. It turns out that the diffusion limit exhibits connections to Prabhakar general fractional calculus. We also analyze in this way a generalization of the space-time fractional Mittag-Leffler process. The approach of construction of good Laplacian generator functions has a large potential in applications of space-time generalizations of the Poisson process and in the field of continuous-time random walks on digraphs.
Keywords
Biased Continuous-time Random Walks, Bernstein matrix functions, Space-time fractional Poisson process, General fractional Calculus, Prabhakar fractional calculus
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.
Comments (0)
We encourage comments and feedback from a broad range of readers. See criteria for comments and our Diversity statement.
Leave a public commentSend a private comment to the author(s)
* All users must log in before leaving a comment