  Diffusive transport of a particle in spatially correlated random energy
landscape having exponential density of states has been considered. We exactly
calculate the diffusivity in the nondispersive quasi-equilibrium transport
regime and found that for slow decaying correlation functions the diffusivity
becomes singular at some particular temperature higher than the temperature of
the transition to the true non-equilibrium dispersive transport regime. It
means that the diffusion becomes anomalous and does not follow the usual
$\propto t^{1/2}$ law. In such situation the fully developed non-equilibrium
regime emerges in two stages: first, at some temperature there is the
transition from the normal to anomalous diffusion, and then at lower
temperature the average velocity for the infinite medium goes to zero, thus
indicating the development of the true dispersive regime. Validity of the
Einstein relation is discussed for the situation where the diffusivity does
exist.
