Origin of the fractional derivative and fractional non-Markovian continuous-time processes

P. Van Mieghem*

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

2 Citations (Scopus)
25 Downloads (Pure)

Abstract

A complex fractional derivative can be derived by formally extending the integer k in the kth derivative of a function, computed via Cauchy's integral, to complex α. This straightforward approach reveals fundamental problems due to inherent nonanalyticity. A consequence is that the complex fractional derivative is not uniquely defined. We explain in detail the anomalies (not closed paths, branch cut jumps) and try to interpret their meaning physically in terms of entropy, friction and deviations from ideal vector fields. Next, we present a class of non-Markovian continuous-time processes by replacing the standard derivative by a Caputo fractional derivative in the classical Chapman-Kolmogorov governing equation of a continuous-time Markov process. The fractional derivative leads to a replacement of the set of exponential base functions by a set of Mittag-Leffler functions, but also creates a complicated dependence structure between states. This fractional non-Markovian process may be applied to generalize the Markovian SIS epidemic process on a contact graph to a more realistic setting.

Original languageEnglish
Article number023242
Number of pages18
JournalPhysical Review Research
Volume4
Issue number2
DOIs
Publication statusPublished - 2022

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