Distributed-order (q,τ)-deformed Lévy processes and their spectral properties - Aldawish, Ibtisam; Ibrahim, Rabha W. (Frontiers in Physics, 2025)
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.