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We derive the probability density of a diffusion process generated bynonergodic velocity fluctuations in presence of a weak potential, using theLiouville equation approach. The velocity of the diffusing particle undergoesdichotomic fluctuations with a given distribution $\psi(\tau)$ of residencetimes in each velocity state. We obtain analytical solutions for the diffusionprocess in a generic external potential and for a generic statistics ofresidence times, including the non-ergodic regime in which the mean residencetime diverges. We show that these analytical solutions are in agreement withnumerical simulations.

Weakly driven anomalous diffusion in non-ergodic regime: an analytical solution