Fractional Compound Poisson Random Fields on Plane

TL;DR

This paper explores fractional variants of compound Poisson random fields on the plane, deriving their distributions, differential equations, and convergence properties, and introduces new fractional Poisson random fields with space and time fractional features.

Contribution

It introduces and analyzes fractional variants of Poisson random fields, including their distributions, differential equations, and convergence, expanding the understanding of fractional stochastic processes.

Findings

Derived distributions of fractional Poisson random fields.

Obtained governing differential equations for specific cases.

Established weak convergence results for the fractional fields.

Abstract

We study a compound Poisson random field on plane and examine its various fractional variants. We derive the distributions of these random fields and in some particular cases, obtain their associated system of governing differential equations. Additionally, various weak convergence results are derived. Also, we introduce and study some fractional variants of the Poisson random field (PRF) that includes the space fractional and the space-time fractional Poisson random fields. We discuss various distributional properties for the fractional PRFs and obtain their time-changed representations. Lastly, we define and study some fractional compound Poisson random fields using the fractional variants of PRF.