Para-Markov chains and related non-local equations

TL;DR

This paper introduces a new class of non-Markovian chains with dependent Mittag-Leffler waiting times, extending fractional Poisson processes and linking them to non-local equations with long memory effects.

Contribution

It defines non-Markovian chains with stochastically dependent Mittag-Leffler waiting times, broadening the scope beyond semi-Markov processes and exploring their connection to non-local equations.

Findings

Developed a new class of non-Markovian chains with dependent waiting times

Extended fractional Poisson process to include dependence in waiting times

Linked these processes to non-local equations with long memory tails

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 our chains, we study a particular counting process which extends the well-known fractional Poisson process, the last one having independent, Mittag-Leffler waiting times.