TY - JOUR
T1 - Origin of the fractional derivative and fractional non-Markovian continuous-time processes
AU - Van Mieghem, P.
PY - 2022
Y1 - 2022
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85133970588&partnerID=8YFLogxK
U2 - 10.1103/PhysRevResearch.4.023242
DO - 10.1103/PhysRevResearch.4.023242
M3 - Article
AN - SCOPUS:85133970588
VL - 4
JO - Physical Review Research
JF - Physical Review Research
SN - 2643-1564
IS - 2
M1 - 023242
ER -