# Para-Markov chains and related non-local equations

## TL;DR

This paper introduces new types of non-Markovian chains with dependent waiting times, leading to long memory effects and extending the fractional Poisson process.

## Contribution

The paper defines new non-Markovian chains with stochastically dependent Mittag-Leffler waiting times, extending beyond semi-Markov models.

## Key findings

- Non-Markovian chains with dependent waiting times create long memory tails in the process evolution.
- A new counting process is introduced that generalizes the fractional Poisson process.
- The new model exhibits different behavior compared to traditional semi-Markov processes.

## Abstract

There is a well-established theory that links semi-Markov chains having Mittag-Leffler waiting times to time-fractional equations. We here go beyond the semi-Markov setting, by defining some non-Markovian chains whose waiting times, although marginally Mittag-Leffler, are assumed to be stochastically dependent. This creates a long memory tail in the evolution, unlike what happens for semi-Markov processes. As a special case of these chains, we study a particular counting process which extends the well-known fractional Poisson process, the last one having independent, Mittag-Leffler waiting times.