α-Stable Lévy Process Overview

• α-stable Lévy processes are continuous stochastic processes with stationary and independent increments characterized by stable laws and heavy tails.
• They exhibit self-similarity, infinite divisibility, and, for α < 2, infinite variance with complex sample path behavior.
• These processes underpin applications such as anomalous diffusion modeling, nonlocal PDE analysis, and robust state-space inference methods.

An α-stable Lévy process is a stochastically continuous process with stationary and independent increments whose finite-dimensional distributions are stable laws with a stability index α ∈ (0,2]. The marginal distributions exhibit infinite divisibility, heavy tails, and, except when α = 2 (Brownian motion), infinite variance. α-stable Lévy processes play fundamental roles in probability theory and statistical physics, modeling phenomena with heavy-tailed increments, anomalous diffusion, and robust long-range dependence. Their deep connections with fractional Laplacian operators, stochastic integrals, statistical inference, and non-Gaussian random fields underlie a broad range of mathematical and applied research.

1. Definition, Characteristic Functions, and Lévy-Khintchine Representation

A real-valued process $L = \{L_{t}\}_{t \ge 0}$ is strictly α-stable with α ∈ (0,2) if for any c > 0

$$\{L_{ct}\}_{t \ge 0} \stackrel{\mathrm{law}}{=} \{c^{1/\alpha} L_t\}_{t \ge 0}, \quad L_0 = 0,$$

and the process has independent, stationary increments (Hintze et al., 2012). The one-dimensional characteristic function takes the form

$\E\bigl[ e^{i\theta L_t} \bigr] = \exp\big(-tC|\theta|^\alpha (1 - i\beta\,\mathrm{sgn}(\theta)\tan(\pi\alpha/2)) + i\mu\theta t \big)$

with parameters α (stability index), β (skewness, in [–1,1]), C (scale > 0), and μ (location in ℝ). For α = 2, this recovers Brownian motion. The Lévy-Khintchine form is

$\E\big[e^{iu L_t}\big] = \exp\big( t\,\Psi(u) \big), \quad \Psi(u) = i\mu u - \tfrac{1}{2} \sigma^2 u^2 + \int_{\mathbb{R} \setminus \{0\}} \big( e^{iuy} - 1 - iuy \mathbf{1}_{|y| \le 1} \big) \nu(dy),$

where ν(dy) = C|y|^{-(1+\alpha)}dy (plus potential skewness) for the strictly α-stable, non-Gaussian case (Hintze et al., 2012, Lin et al., 2011). The process has càdlàg paths with jumps of all sizes, with no continuous modification unless α = 2.

2. Fundamental Properties and Domains of Attraction

Key properties include:

• Infinite divisibility: Every α-stable law arises as the infinite divisible limit of sums of i.i.d. random variables.
• Heavy tails:

$\Pr\{|L_t| > x\} \sim kx^{-\alpha}$ as $x \to \infty$.

• Self-similarity: $L_{ct} \overset{d}{=} c^{1/\alpha}L_t$ for all c > 0.
• Infinite variance: For α < 2, variance diverges; only for α = 2 (Gaussian law) is variance finite.
• Sample path regularity: Càdlàg (right-continuous with left limits), almost surely an infinite number of jumps in every finite interval (Godsill et al., 2019, Feltes et al., 2020).

A real i.i.d. sequence $\{X_j\}$ is in the domain of attraction of an α-stable law if there exists norming $b_n$ such that $S_n / b_n \Rightarrow Z_\alpha(1)$, with $$\{L_{ct}\}_{t \ge 0} \stackrel{\mathrm{law}}{=} \{c^{1/\alpha} L_t\}_{t \ge 0}, \quad L_0 = 0,$$0 and slowly varying L (Lin et al., 2011).

3. Generators, Fractional Laplacian, and Multivariate Extension

For an isotropic symmetric α-stable Lévy process in $$\{L_{ct}\}_{t \ge 0} \stackrel{\mathrm{law}}{=} \{c^{1/\alpha} L_t\}_{t \ge 0}, \quad L_0 = 0,$$1, the characteristic exponent is $$\{L_{ct}\}_{t \ge 0} \stackrel{\mathrm{law}}{=} \{c^{1/\alpha} L_t\}_{t \ge 0}, \quad L_0 = 0,$$2 and the Lévy measure is $$\{L_{ct}\}_{t \ge 0} \stackrel{\mathrm{law}}{=} \{c^{1/\alpha} L_t\}_{t \ge 0}, \quad L_0 = 0,$$3 (Guo et al., 30 Jan 2026, Bogdan et al., 2024): $$\{L_{ct}\}_{t \ge 0} \stackrel{\mathrm{law}}{=} \{c^{1/\alpha} L_t\}_{t \ge 0}, \quad L_0 = 0,$$4 The fractional Laplacian is

$$\{L_{ct}\}_{t \ge 0} \stackrel{\mathrm{law}}{=} \{c^{1/\alpha} L_t\}_{t \ge 0}, \quad L_0 = 0,$$5

This operator generates the semigroup of the α-stable Lévy process and links α-stable motions to fractional PDEs, nonlocal Dirichlet forms, and many physical models (Guo et al., 30 Jan 2026, Zhang et al., 2020).

4. Weak Convergence, Stochastic Integrals, and Functional Limit Theorems

The finite-dimensional convergence of α-stable processes under scaling and the corresponding functional central limit theorems are established in both finite and infinite-dimensional settings (Balan et al., 2018, Scalas et al., 2013). Generalizations always rely on the regular variation and infinite divisibility of increments:

• For any continuous and bounded function f,

$$\{L_{ct}\}_{t \ge 0} \stackrel{\mathrm{law}}{=} \{c^{1/\alpha} L_t\}_{t \ge 0}, \quad L_0 = 0,$$6

where $$\{L_{ct}\}_{t \ge 0} \stackrel{\mathrm{law}}{=} \{c^{1/\alpha} L_t\}_{t \ge 0}, \quad L_0 = 0,$$7 is symmetric α-stable and $$\{L_{ct}\}_{t \ge 0} \stackrel{\mathrm{law}}{=} \{c^{1/\alpha} L_t\}_{t \ge 0}, \quad L_0 = 0,$$8 is an inverse subordinator (Scalas et al., 2013).

• For partial sums of regularly varying elements in Skorokhod space,

$$\{L_{ct}\}_{t \ge 0} \stackrel{\mathrm{law}}{=} \{c^{1/\alpha} L_t\}_{t \ge 0}, \quad L_0 = 0,$$9

with Z an infinite-dimensional α-stable Lévy motion (Balan et al., 2018).

The stochastic integral driven by an α-stable Lévy process takes the form (Lin et al., 2011): $\E\bigl[ e^{i\theta L_t} \bigr] = \exp\big(-tC|\theta|^\alpha (1 - i\beta\,\mathrm{sgn}(\theta)\tan(\pi\alpha/2)) + i\mu\theta t \big)$0 Its definition leverages the semimartingale decomposition via Poisson random measures and compensators.

5. State-Space Models, Inference, and Simulation

α-stable Lévy processes are employed in state-space models due to their ability to encode non-Gaussian heavy-tailed noise, both in finite- and infinite-dimensional SDE systems. The shot-noise (LePage) representation is central: $\E\bigl[ e^{i\theta L_t} \bigr] = \exp\big(-tC|\theta|^\alpha (1 - i\beta\,\mathrm{sgn}(\theta)\tan(\pi\alpha/2)) + i\mu\theta t \big)$1 with $\E\bigl[ e^{i\theta L_t} \bigr] = \exp\big(-tC|\theta|^\alpha (1 - i\beta\,\mathrm{sgn}(\theta)\tan(\pi\alpha/2)) + i\mu\theta t \big)$2 Poisson arrival times, $\E\bigl[ e^{i\theta L_t} \bigr] = \exp\big(-tC|\theta|^\alpha (1 - i\beta\,\mathrm{sgn}(\theta)\tan(\pi\alpha/2)) + i\mu\theta t \big)$3 marks, and $\E\bigl[ e^{i\theta L_t} \bigr] = \exp\big(-tC|\theta|^\alpha (1 - i\beta\,\mathrm{sgn}(\theta)\tan(\pi\alpha/2)) + i\mu\theta t \big)$4 uniform times. This yields conditionally Gaussian constructions and tractable building blocks for Rao-Blackwellized sequential Monte Carlo inference, enabling analytic marginalizations over scale and skewness parameters (Godsill et al., 2019).

Monte Carlo and neural sampling schemes—such as fractional walk-on-spheres (FWoS) and fractional neural walk-on-spheres (FNWoS)—efficiently solve problems featuring α-stable drivers by exploiting their probabilistic and path properties (Guo et al., 30 Jan 2026).

6. Reflected, Locally Stable, and Non-Standard α-Stable Processes

Reflecting α-stable Lévy processes in bounded domains yields reflected jump processes, with explicit constructions via nonlocal Schrödinger perturbations, supermedian functions, and ladder process semigroups, maintaining strong Markov and ergodicity properties (Bogdan et al., 2024).

Locally α-stable Lévy-type processes allow spatially varying jump intensities and skewness, preserving the essential scaling and infinite divisibility locally. In law, such processes can be approximated by nonlinear regressions: $\E\bigl[ e^{i\theta L_t} \bigr] = \exp\big(-tC|\theta|^\alpha (1 - i\beta\,\mathrm{sgn}(\theta)\tan(\pi\alpha/2)) + i\mu\theta t \big)$5 with explicit error estimates in total variation and uniform topology (Kulik, 2018).

7. Applications: Anomalous Transport, Exit Problems, and High-Dimensional Analysis

The α-stable Lévy process provides the canonical model for Lévy flights and anomalous transport, characterized by heavy-tailed, non-Gaussian jumps. The exit time problem, fundamental for anomalous diffusion, is addressed using probabilistic numerical algorithms that approximate the process as a mixture of Brownian motion and compound Poisson jumps, yielding efficient schemes for high-dimensional bounded domains (Yang et al., 14 Jan 2026).

Monte Carlo and “walk-on-half-spaces” algorithms simulate first-entry distributions into spatial slabs, leveraging explicit n-tuple fluctuation identities and orthogonal coordinate decompositions (Kyprianou et al., 2024). In functional data and random fields, α-stable Lévy motions in Skorokhod spaces permit limit theorems for sums of heavy-tailed random functions (Balan et al., 2018).

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