AUTHOR=Aldawish Ibtisam , Ibrahim Rabha W. 

TITLE=Distributed-order (q,τ)-deformed Lévy processes and their spectral properties

JOURNAL=Frontiers in Physics

VOLUME=Volume 13 - 2025

YEAR=2025

URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2025.1647182

DOI=10.3389/fphy.2025.1647182

ISSN=2296-424X

ABSTRACT=Lévy processes play a central role in stochastic modeling, providing a unifying framework for jump dynamics, anomalous diffusion, and heavy-tailed phenomena across physics and applied sciences. We propose a novel framework for (q,τ,α,β)-generalized Lévy processes, extending fractional and tempered stable models with (q,τ)-Gamma and (q,τ)-Mittag--Leffler functions. The construction uses Laplace transforms of (q,τ)-inverse subordinators combined with the Lévy--Khintchine representation to obtain explicit expressions for characteristic functions. Numerical results show how variations in q and τ affect Γq,τ(x) and Eβ(q,τ)(z), leading to slower relaxation, heavier tails, and enhanced memory effects relative to classical counterparts. These outcomes demonstrate that (q,τ)-deformation provides a flexible mechanism for modeling anomalous diffusion, nonlocal dynamics, and heavy-tailed processes relevant in physics, finance, and geophysics.