Characterization of diffusion regimes in systems with memory and external noise

In this work we investigate the diffusion of a particle governed by a generalized Langevin equation subjected to active external stochastic forces. We derive the time-dependent diffusion coefficient and a generalized form of the Kubo relation that accounts for the presence of external noise, offering a broader framework for analyzing nonequilibrium stochastic systems. Based on this result, we obtain a simple and general criterion to determine whether the particle exhibits normal diffusion, subdiffusion, or superdiffusion. Our approach enables the determination of the asymptotic behavior of the diffusion coefficient based solely on the properties of the memory kernel and external noise. To illustrate the utility of this method, we apply it to three representative cases, obtaining the long-time behavior of the mean-square displacement, and from the diffusion exponent we get the diffusion regime diagram, including remarkably a region of fast superdiffusion.