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

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