Generalized Space-Fractional Poisson Process via Variable-Order Stable Subordinator
Reetendra Singh, Aditya Maheshwari
TL;DR
This work generalizes the space-fractional Poisson process by introducing a variable-order stable subordinator $S^{α(t)}(t)$ with $α(t)\in(0,1)$ and constructing the Generalized Space-Fractional Poisson Process via Variable-Order Subordinator (GSFPP-VO) as $\{N(S^{α(t)}(t))\}_{t\ge0}$. By treating $S^{α(t)}(t)$ as the time change of a homogeneous Poisson process $N(t)$ with rate $λ>0$, the authors derive explicit forms for the Laplace transform, pgf, pmf, and mgf of the GSFPP-VO and show these satisfy associated PDEs, while also developing the generalized distributions and hitting-time analysis. The time-varying framework is built from a partition of time with constant local orders, yielding a Laplace representation $\mathbb{E}[e^{-λ S^{α(t)}(t)}]=e^{-\ int_0^{t} λ^{α(s)} ds}$ and a pgf $ψ_{α(t)}(u,t)=e^{-\ int_0^{t} λ^{α(s)} (1-u)^{α(s)} ds}$, which facilitate closed-form expressions for pmf, mgf, and hitting-time probabilities, along with an explicit Lévy measure characterization. These results extend SFPP theory to a time-varying order, enabling flexible modeling of count processes with evolving fractional dynamics and heavy-tailed behavior across applications. The paper thus provides a cohesive analytic framework for GSFPP-VO, including transformations, distributions, hitting times, and jump structure, with potential implications for areas requiring nonstationary fractional counting models.
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.