Superdiffusion in stochastic processes with long-range memory

We investigate non-Markovian stochastic processes with long-range temporal memory described by a generalized Langevin (Mori-Zwanzig) equation with a power-law memory kernel. Special attention is paid to the critical regime where the system parameters lie on the boundary separating stable and unstable dynamics, leading to diffusionlike behavior with memory. We derive analytical expressions for the asymptotic behavior of the process and show that, under appropriate conditions on the memory kernel, the system exhibits superdiffusive scaling. In particular, we establish an explicit relation between the exponent of the power-law memory kernel and the growth rate of the variance. Analytical predictions are supported by numerical simulations, confirming the emergence of superdiffusive regimes in systems with long-range memory.