Some Spatial Point Processes of Poisson Family

TL;DR

This paper introduces and analyzes new spatial Poisson-type point processes, including the generalized Poisson random field and Skellam-type processes, exploring their properties, fractional variants, and differential equations.

Contribution

It presents novel Poisson-type processes like the GPRF and their fractional variants, expanding the mathematical modeling toolkit for spatial point patterns.

Findings

Distributional properties derived

Governing differential equations established

Fractional variants analyzed in detail

Abstract

Spatial Poisson point processes on finite-dimensional Euclidean space provide fundamental mathematical tools for modeling random spatial point patterns. In this paper, we introduce and analyze several Poisson-type spatial point processes. In particular, we propose and study a point process, namely, the generalized Poisson random field (GPRF), in which more than one point can be observed with positive probability, within a rectangular region having infinitesimal Lebesgue measure. A thinning of the GPRF into independent GPRFs with reduced rate parameters is discussed. Furthermore, we consider these processes indexed by the positive quadrant of the plane and analyze their fractional variants. Various distributional properties of these processes and related governing differential equations are obtained. Later, we define and analyze a spatial Skellam-type point process via GPRF. Moreover, a fractional variant of it in the two parameter case is studied in detail.