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Open AccessArticle

Biased Continuous-Time Random Walks with Mittag-Leffler Jumps

1
Institut Jean le Rond d’Alembert, Sorbonne Université, CNRS UMR 7190, 4 Place Jussieu, 75252 Paris CEDEX 05, France
2
Department of Mathematics “Giuseppe Peano”, University of Torino, 10123 Torino, Italy
3
Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, Ciudad de México 01000, Mexico
*
Author to whom correspondence should be addressed.
Fractal Fract. 2020, 4(4), 51; https://doi.org/10.3390/fractalfract4040051
Received: 1 October 2020 / Revised: 28 October 2020 / Accepted: 30 October 2020 / Published: 31 October 2020
(This article belongs to the Special Issue Fractional Behavior in Nature 2019)
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 time-changed with an independent (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 Mittag-Leffler process. The approach of constructing 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. View Full-Text
Keywords: space-time generalizations of Poisson process; biased continuous-time random walks; Bernstein functions; Prabhakar fractional calculus space-time generalizations of Poisson process; biased continuous-time random walks; Bernstein functions; Prabhakar fractional calculus
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MDPI and ACS Style

Michelitsch, T.M.; Polito, F.; Riascos, A.P. Biased Continuous-Time Random Walks with Mittag-Leffler Jumps. Fractal Fract. 2020, 4, 51. https://doi.org/10.3390/fractalfract4040051

AMA Style

Michelitsch TM, Polito F, Riascos AP. Biased Continuous-Time Random Walks with Mittag-Leffler Jumps. Fractal and Fractional. 2020; 4(4):51. https://doi.org/10.3390/fractalfract4040051

Chicago/Turabian Style

Michelitsch, Thomas M.; Polito, Federico; Riascos, Alejandro P. 2020. "Biased Continuous-Time Random Walks with Mittag-Leffler Jumps" Fractal Fract. 4, no. 4: 51. https://doi.org/10.3390/fractalfract4040051

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