Generalized Space-Fractional Poisson Process (GSFPP-VO)

• The paper introduces the GSFPP-VO, generalizing classical Poisson processes by time-changing them with variable-order stable subordinators.
• It details explicit transforms, probability generating functions, and discrete Lévy measures to capture non-stationary, heavy-tailed jump dynamics.
• Implications include unified modeling of fractional dynamics and practical applications in systems with abrupt shifts in stochastic behavior.

The Generalized Space-Fractional Poisson Process via @@@@1@@@@ (GSFPP-VO) is a class of time-inhomogeneous integer-valued stochastic processes defined via time-changing a homogeneous Poisson process by an independent variable-order stable subordinator (VOSS). This framework generalizes the classical space-fractional Poisson processes by allowing the stability index to exhibit right-continuous piecewise-constant variability in time, producing rich probabilistic and analytic structures that unify fixed-order and inhomogeneous fractional Poissonian dynamics. The resulting process admits explicit representations for its transforms, discrete Lévy characteristics, and hitting-time distributions, and it serves as a canonical model for variable-order random evolution with non-stationary increments (Singh et al., 11 Jan 2026, Beghin et al., 2016).

1. Construction of the Variable-Order Stable Subordinator

Let $\alpha: [0, \infty) \to (0,1)$ be a right-continuous, piecewise-constant function partitioned by $0 = t_0 < t_1 < \cdots < t_n = t$, where on each interval $[t_{k-1}, t_k)$, $\alpha(s) = \alpha_k$. For each subinterval, $S^{\alpha_k}(\cdot)$ is an independent classical $\alpha_k$-stable subordinator. The Variable-Order Stable Subordinator (VOSS) is defined as:

$$S^{\alpha(\cdot)}(t) = \sum_{k=1}^{n} \Bigl[ S^{\alpha_k}(t_k) - S^{\alpha_k}(t_{k-1}) \Bigr]$$

The increment $S^{\alpha(\cdot)}(r) - S^{\alpha(\cdot)}(s)$ over $[s, r)$ has independent increments corresponding to the overlaps of $[t_{k-1}, t_k)$ with $[s, r)$.

The Laplace transform of $S^{\alpha(\cdot)}(t)$ is

$$\mathbb{E}\left[ e^{-\mu S^{\alpha(\cdot)}(t) } \right] = \exp\left\{ -\int_0^t \mu^{\alpha(s)} ds \right\}$$

This explicitly encodes the variable fractional order along the time axis (Singh et al., 11 Jan 2026).

2. Definition and Distributional Properties of GSFPP-VO

Let $N(t, \lambda)$ denote a homogeneous Poisson process of rate $\lambda>0$, independent of $S^{\alpha(\cdot)}$. The GSFPP-VO is defined as the time-changed process:

$X(t) = N( S^{\alpha(\cdot)}(t), \lambda )$

The probability generating function (PGF) follows by conditioning:

$$G(u, t) = \mathbb{E}[ u^{X(t)} ] = \exp\left\{ - \int_0^t [\lambda(1-u)]^{\alpha(s)} ds \right\}$$

The process is infinitely divisible with state probabilities (pmf) expressible, for piecewise-constant order, as multinomial sums involving Pochhammer symbols:

$$p_k(t) = \sum_{r=0}^{\infty} \frac{(-1)^r}{r!} \sum_{\substack{x_1+\cdots+x_n=r, \, x_i\geq0}} \frac{r!}{x_1! \cdots x_n!} \left( \prod_{i=1}^n [\lambda^{\alpha_i}(t_i-t_{i-1})]^{x_i} \right) \frac{(-1)^k}{k!} (\alpha_1 x_1 + \cdots + \alpha_n x_n)_k$$

where $(\cdot)_k$ denotes the Pochhammer symbol (Singh et al., 11 Jan 2026).

This structure generalizes the fixed-index space-fractional Poisson process described by Orsingher and Polito, reducing to classical forms for constant $\alpha(t)$ (Singh et al., 11 Jan 2026, Beghin et al., 2016).

3. Evolution Equations and Fractional Dynamics

Let $B$ denote the backward shift operator on $k$. The Kolmogorov-type forward equation for the pmf is

$$\frac{d}{dt} p_k^{\alpha(\cdot)}(t) = -\lambda^{\alpha(t)} (1-B)^{\alpha(t)} p_k^{\alpha(\cdot)}(t), \qquad p_k^{\alpha(\cdot)}(0) = \delta_{k,0}$$

with the variable-order fractional difference operator defined by its binomial expansion:

$$(1-B)^{\alpha(t)} = \sum_{j=0}^\infty \binom{\alpha(t)}{j} (-1)^j B^j$$

Alternatively, the PGF $\psi(u,t)$ satisfies the partial differential equation:

$$\frac{\partial}{\partial t} \psi(u,t) = -\lambda^{\alpha(t)} (1-u)^{\alpha(t)} \psi(u,t), \qquad \psi(u,0) = 1$$

This exponential PDE is solved by the PGF form given above. These equations admit direct specialization to the classical and space-fractional Poisson cases for constant $\alpha$ (Singh et al., 11 Jan 2026, Beghin et al., 2016).

4. Lévy-Khintchine Structure and Infinite Divisibility

The GSFPP-VO is, at each $t$, an infinitely divisible integer-valued random variable, with discrete Lévy measure governed by the exponent

$$\Phi_t(u) = -\log \mathbb{E}[ u^{X(t)} ] = \int_0^t \lambda^{\alpha(s)} (1-u)^{\alpha(s)} ds$$

Expanding $(1-u)^{\alpha(s)}$ yields the Lèvy intensity:

$$\nu_t(m) = \int_0^t \lambda^{\alpha(s)} (-1)^{m+1} \binom{\alpha(s)}{m} ds, \qquad m=1,2,\ldots$$

where $$\binom{\alpha}{m} = \frac{\alpha(\alpha-1)\cdots(\alpha-m+1)}{m!}$$. The rate at which $m$ identical jumps occur in an infinitesimal time $ds$ is $$\lambda^{\alpha(s)} (-1)^{m+1} \binom{\alpha(s)}{m} ds$$. This structure is distinct from the usual Lévy-Khintchine exponents for continuous-variable processes, emphasizing the discrete, jump-driven nature of GSFPP-VO (Singh et al., 11 Jan 2026).

5. Hitting-Time Distributions

The $k$th hitting time is defined by

$\tau_k = \inf\{ t \geq 0 : X(t) \geq k \}$

The cumulative distribution function is

$$\mathbb{P}\{\tau_k < t\} = \sum_{m=k}^\infty p_m(t) = 1 - \sum_{m=0}^{k-1} p_m(t)$$

Differentiation (in the case of piecewise-smooth $\alpha(\cdot)$) provides the hitting-time density. The hitting-time distribution inherits the complexity and forced non-Markovianity of the underlying process, and the explicit multinomial expressions mirror those of the pmf (Singh et al., 11 Jan 2026).

6. Special Cases and Limiting Regimes

Several parametric regimes recover known models:

• Constant $\alpha(t)\equiv\alpha$ yields the classical space-fractional Poisson process of Orsingher–Polito, with PGF $G(u,t) = \exp\{ -\lambda^\alpha t (1-u)^\alpha \}$ and established series forms for the pmf.
• $\alpha=1$ reduces to the standard Poisson process, as the stable subordinator becomes deterministic: $S^1(t) = t$, $X(t) = N(t)$.
• Piecewise-constant choices (e.g., two-state regimes) allow explicit modeling of abrupt shifts between different fractional dynamics.
• If $\alpha \to 0$, the process degenerates, with $G(u,t) \to \exp(-t)$ and the entire probability mass eventually concentrated at zero, indicating process extinction (this requires interpretive care, as infinite-activity subordinators cease to be well-defined) (Singh et al., 11 Jan 2026, Beghin et al., 2016).

7. Connections and Underlying Significance

The GSFPP-VO emerges as a canonical example of fractional-in-time random evolution with variable-order characteristics, broadening the scope of both fractional Poisson models and time-changed Lévy processes. It underpins modeling frameworks for non-homogeneous, anomalous, or aging environments in applied probability, with explicit analytical forms supporting both theoretical study and computational practice. The broad class absorbs and extends classical models as special cases, establishing a bridge to non-stationary stochastic modeling of systems with temporal heterogeneity in their jump or waiting-time statistics (Singh et al., 11 Jan 2026, Beghin et al., 2016).

Process	Subordinator	Fractional index	Limiting behavior
Standard Poisson	Deterministic, $t$	$\alpha\to1$	Markovian, stationary
Space-frac. Poisson (O-P)	Stable, fixed	$0<\alpha<1$	Heavy-tailed jumps
GSFPP-VO	VOSS, piecewise	variable	Non-stationary, inhomogeneous

The construction and explicit calculations associated with the GSFPP-VO provide a foundation for further exploration of random measures with variable-order self-similarity, non-homogeneous noise models, and fractional-order differential equations with time-dependent parameters (Singh et al., 11 Jan 2026, Beghin et al., 2016).