  Many biological processes involve one dimensional diffusion over a correlated
inhomogeneous energy landscape with a correlation length $\xi_c$. Typical
examples are specific protein target location on DNA, nucleosome repositioning,
or DNA translocation through a nanopore, in all cases with $\xi_c\approx$ 10
nm. We investigate such transport processes by the mean first passage time
(MFPT) formalism, and find diffusion times which exhibit strong sample to
sample fluctuations. For a a displacement $N$, the average MFPT is diffusive,
while its standard deviation over the ensemble of energy profiles scales as
$N^{3/2}$ with a large prefactor. Fluctuations are thus dominant for
displacements smaller than a characteristic $N_c \gg \xi_c$: typical values are
much less than the mean, and governed by an anomalous diffusion rule. Potential
biological consequences of such random walks, composed of rapid scans in the
vicinity of favorable energy valleys and occasional jumps to further valleys,
is discussed.
