Biased Continuous-Time Random Walks with Mittag-Leffler Jumps
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...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-10-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/4/4/51 |
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| author | Thomas M. Michelitsch Federico Polito Alejandro P. Riascos |
| author_facet | Thomas M. Michelitsch Federico Polito Alejandro P. Riascos |
| author_sort | Thomas M. Michelitsch |
| collection | DOAJ |
| description | 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 ‘<i>space-time Mittag-Leffler process</i>’. 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. |
| first_indexed | 2024-03-10T15:10:15Z |
| format | Article |
| id | doaj.art-1e17de1aa36d41b28c6b21ea5e734c73 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-10T15:10:15Z |
| publishDate | 2020-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-1e17de1aa36d41b28c6b21ea5e734c732023-11-20T19:24:29ZengMDPI AGFractal and Fractional2504-31102020-10-01445110.3390/fractalfract4040051Biased Continuous-Time Random Walks with Mittag-Leffler JumpsThomas M. Michelitsch0Federico Polito1Alejandro P. Riascos2Institut Jean le Rond d’Alembert, Sorbonne Université, CNRS UMR 7190, 4 Place Jussieu, 75252 Paris CEDEX 05, FranceDepartment of Mathematics “Giuseppe Peano”, University of Torino, 10123 Torino, ItalyInstituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, Ciudad de México 01000, MexicoWe 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 ‘<i>space-time Mittag-Leffler process</i>’. 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.https://www.mdpi.com/2504-3110/4/4/51space-time generalizations of Poisson processbiased continuous-time random walksBernstein functionsPrabhakar fractional calculus |
| spellingShingle | Thomas M. Michelitsch Federico Polito Alejandro P. Riascos Biased Continuous-Time Random Walks with Mittag-Leffler Jumps Fractal and Fractional space-time generalizations of Poisson process biased continuous-time random walks Bernstein functions Prabhakar fractional calculus |
| title | Biased Continuous-Time Random Walks with Mittag-Leffler Jumps |
| title_full | Biased Continuous-Time Random Walks with Mittag-Leffler Jumps |
| title_fullStr | Biased Continuous-Time Random Walks with Mittag-Leffler Jumps |
| title_full_unstemmed | Biased Continuous-Time Random Walks with Mittag-Leffler Jumps |
| title_short | Biased Continuous-Time Random Walks with Mittag-Leffler Jumps |
| title_sort | biased continuous time random walks with mittag leffler jumps |
| topic | space-time generalizations of Poisson process biased continuous-time random walks Bernstein functions Prabhakar fractional calculus |
| url | https://www.mdpi.com/2504-3110/4/4/51 |
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