Anomalous random flights and time-fractional run-and-tumble equations

TL;DR

This paper extends the analysis of random flights to include time-fractional processes, revealing anomalous behaviors and providing insights into complex systems in physics and biology through a probabilistic and stochastic framework.

Contribution

It introduces a non-local generalization of kinetic equations for random flights, analyzing the time-fractional telegraph process and its stochastic solutions.

Findings

Time-fractional derivatives induce anomalous transport behaviors.

The stochastic solutions are interpreted as time-changed random processes.

Results applicable to complex systems in physics and biology.

Abstract

Random flights (also called run-and-tumble walks or transport processes) represent finite velocity random motions changing direction at any Poissonian time. These models in d-dimension, can be studied giving a general formulation of the problem valid at any spatial dimension. The aim of this paper is to extend this general analysis to time-fractional processes arising from a non-local generalization of the kinetic equations. The probabilistic interpretation of the solution of the time-fractional equations leads to a time-changed version of the original transport processes. The obtained results provides a clear picture of the role played by the time-fractional derivatives in this kind of random motions. They display an anomalous behavior and are useful to describe several complex systems arising in statistical physics and biology. In particular, we focus on the one-dimensional random flight, called telegraph process, studying the time-fractional version of the classical telegraph equation and providing a suitable interpretation of its stochastic solutions.