Fractional Poisson Processes of Order k and Beyond

TL;DR

This paper introduces fractional Poisson processes of order k in multiple dimensions, generalizing classical Poisson processes with heavy-tailed waiting times and infinite arrivals, along with their governing equations and martingale properties.

Contribution

It defines new fractional Poisson processes of order k in Euclidean space, deriving their probabilities, differential equations, and martingale characterizations, extending classical models.

Findings

Derived marginal probabilities and difference-differential equations.

Established Watanabe martingale characterization.

Showed processes generalize classical Poisson models with heavy tails.

Abstract

In this article, we introduce fractional Poisson felds of order k in n-dimensional Euclidean space $R_n^+$. We also work on time-fractional Poisson process of order k, space-fractional Poisson process of order k and tempered version of time-space fractional Poisson process of order k in one dimensional Euclidean space $R_1^+$. These processes are defined in terms of fractional compound Poisson processes. Time-fractional Poisson process of order k naturally generalizes the Poisson process and Poisson process of order k to a heavy tailed waiting times counting process. The space-fractional Poisson process of order k, allows on average infinite number of arrivals in any interval. We derive the marginal probabilities, governing difference-differential equations of the introduced processes. We also provide Watanabe martingale characterization for some time-changed Poisson processes.