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.
Aldawish et al. (Mon,) studied this question.