Fractional processes: from Poisson to branching one

TL;DR

This paper explores fractional generalizations of Poisson and branching processes, linking them to fractional differential equations and Levy stable densities, and develops Monte Carlo algorithms for their simulation.

Contribution

It introduces fractional versions of Poisson and branching processes, connecting them to fractional calculus and Levy densities, and provides simulation methods and limit distribution analysis.

Findings

Derived limit distributions for normalized variables in fractional processes.

Developed Monte Carlo algorithms for simulating fractional processes.

Demonstrated numerical calculations confirming theoretical results.

Abstract

Fractional generalizations of the Poisson process and branching Furry process are considered. The link between characteristics of the processes, fractional differential equations and Levy stable densities are discussed and used for construction of the Monte Carlo algorithm for simulation of random waiting times in fractional processes. Numerical calculations are performed and limit distributions of the normalized variable Z=N/<N> are found for both processes.