Generalized Space-Fractional Poisson Process via Variable-Order Stable Subordinator

TL;DR

This paper introduces a new generalized space-fractional Poisson process driven by a variable-order stable subordinator, providing explicit formulas and analyzing its properties, including distributions, hitting times, and Lévy measures.

Contribution

It develops the GSFPP-VO model using a novel variable-order stable subordinator, deriving explicit formulas and PDEs, and analyzing its probabilistic properties.

Findings

Explicit Laplace transform and generating functions derived.

Distribution and hitting-time properties analyzed.

Lévy measures characterized.

Abstract

This paper introduces a variable-order stable subordinator (VOSS) $S^{\alpha(t)}(t)$ with index $\alpha(t)\in(0,1)$, where $\alpha(t)$ is a right-continuous piecewise constant function. We drive the Generalized Space-Fractional Poisson Process via Variable-Order Stable Subordinator (GSFPP-VO) defined by $\{N(S^{\alpha(t)}(t))\}_{t \geq 0}$, obtained by time-changing a homogeneous Poisson process $\{N(t,\lambda)\}_{t\geq 0}$ with rate parameter $\lambda>0$ by an independent VOSS. Explicit expressions for the Laplace transform, probability generating function, probability mass function, and moment generating function of the GSFPP-VO are derived, and these quantities are shown to satisfy partial differential equations. Finally, we establish the associated generalized distributions, analyze the hitting-time properties, and characterize the L\'evy measures of the GSFPP-VO.