Distributed-Order (q, τ)-Deformed Lévy Processes and Their Spectral Properties

Ibtisam Aldawish¹Rabha W. Ibrahim^2,3*

• ¹Imam Muhammad Ibn Saud Islamic University, Riyadh, Saudi Arabia
• ²Institute of Electrical and Electronics Engineers, New York, United States
• ³Alayen Iraqi University, Nasiriyah, Iraq

This paper investigates the construction and analysis of distributedorder (q, τ)-deformed Lévy processes and their corresponding nonlocal fractional generators. By incorporating the (q, τ)-Gamma and (q, τ)-Mittag-Leffler functions into the Lévy-Khintchine framework, we derive generalized generators that exhibit tunable memory and multi-scale behavior. We formally establish the existence of these processes, characterize the infinitesimal generators for both fixed-order and distributed-order models, and provide explicit spectral representations. In particular, we analyze the scaling coefficient K (α,β) q,τ governing the behavior of the generator in Fourier space, and compare its asymptotic approximation with exact numerical results across a range of deformation parameters (q, τ). The study reveals how the (q, τ) deformation parameters and distributed fractional order introduce rich flexibility in modeling anomalous diffusion and complex transport phenomena, with potential applications in physics, finance, and biology. Numerical examples and detailed comparison figures are presented to validate the theoretical results and to illustrate the spectral impact of the deformation parameters.