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

Abstract

This paper introduces a variable-order stable subordinator (VOSS) $S^{\hat{I}\pm (t)}(t)$ with index $\hat{I}\pm (t)\in (0,1)$, where $\hat{I}\pm (t)$ is a right-continuous piecewise constant function. We drive the Generalized Space-Fractional Poisson Process via Variable-Order Stable Subordinator (GSFPP-VO) defined by $\{N(S^{\hat{I}\pm (t)}(t)){\}}_{t\ge 0}$, obtained by time-changing a homogeneous Poisson process $\{N(t,\hat{I}»){\}}_{t\ge 0}$ with rate parameter $\hat{I}»>0$ 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 Lévy measures of the GSFPP-VO.

9 pages