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Isolation and Culture of Primary Mouse Keratinocytes from Neonatal and Adult Mouse Skin

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Skellam Type Processes of Order k and Beyond.

Neha Gupta¹, Arun Kumar¹, Nikolai Leonenko²

¹Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar, Punjab 140001, India.

Entropy (Basel, Switzerland)

|December 8, 2020

Summary

We introduce the Skellam process of order k and its time-changed variants, including space-fractional and tempered versions. These novel processes generalize existing models and offer new insights into stochastic processes.

Keywords:

Lévy measure Poisson process of order k Skellam process running average subordination

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Area of Science:

Probability theory
Stochastic processes
Mathematical analysis

Background:

The Skellam distribution models the difference between two independent Poisson random variables.
Generalizations of the Skellam process are essential for modeling complex phenomena.

Purpose of the Study:

Introduce the Skellam process of order k and its running average.
Investigate time-changed Skellam processes, including space-fractional and tempered versions.
Derive key properties of these novel stochastic processes.

Main Methods:

Development of the Skellam process of order k and its running average.
Application of time changes using independent stable and tempered stable subordinators.
Derivation of marginal probabilities, Lévy measures, and governing equations.

Main Results:

Introduction of the Skellam process of order k and its running average.
Characterization of time-changed Skellam processes, including space-fractional and tempered variants.
Derivation of marginal distributions, Lévy measures, and difference-differential equations.

Conclusions:

The introduced processes generalize the classical Skellam process and the running average of the Poisson process.
The study provides a comprehensive analysis of the properties of these novel stochastic models.
This work offers a foundation for further research in fractional and tempered stochastic processes.