# Fractional compound Poisson processes with multiple internal states

## TL;DR

This paper develops mathematical models for particles exhibiting anomalous diffusion with multiple internal states, deriving equations to describe their positions, internal state distributions, and related functionals, with applications in first passage and occupation times.

## Contribution

It introduces new fractional Fokker-Planck and Feynman-Kac equations for multi-state anomalous diffusion processes, including internal state dynamics.

## Key findings

- Derived equations for particle positions and internal states
- Analyzed stochastic process dynamics
- Applied models to first passage and occupation time distributions

## Abstract

For the particles undergoing the anomalous diffusion with different waiting time distributions for different internal states, we derive the Fokker-Planck and Feymann-Kac equations, respectively, describing positions of the particles and functional distributions of the trajectories of particles; in particular, the equations governing the functional distribution of internal states are also obtained. The dynamics of the stochastic processes are analyzed and the applications, calculating the distribution of the first passage time and the distribution of the fraction of the occupation time, of the equations are given.