  We derive the probability density of a diffusion process generated by
nonergodic velocity fluctuations in presence of a weak potential, using the
Liouville equation approach. The velocity of the diffusing particle undergoes
dichotomic fluctuations with a given distribution $\psi(\tau)$ of residence
times in each velocity state. We obtain analytical solutions for the diffusion
process in a generic external potential and for a generic statistics of
residence times, including the non-ergodic regime in which the mean residence
time diverges. We show that these analytical solutions are in agreement with
numerical simulations.
