  We study the dynamics of a Brownian particle in a strongly correlated
quenched random potential defined as a periodically-extended (with period $L$)
finite trajectory of a fractional Brownian motion with arbitrary Hurst exponent
$H \in (0,1)$. While the periodicity ensures that the ultimate long-time
behavior is diffusive, the generalised Sinai potential considered here leads to
a strong logarithmic confinement of particle trajectories at intermediate
times. These two competing trends lead to dynamical frustration and result in a
rich statistical behavior of the diffusion coefficient $D_L$: Although one has
the typical value $D^{\rm typ}_L \sim \exp(-\beta L^H)$, we show via an exact
analytical approach that the positive moments ($k>0$) scale like $\langle D^k_L
\rangle \sim \exp{[-c' (k \beta L^{H})^{1/(1+H)}]}$, and the negative ones as
$\langle D^{-k}_L \rangle \sim \exp(a' (k \beta L^{H})^2)$, $c'$ and $a'$ being
numerical constants and $\beta$ the inverse temperature. These results
demonstrate that $D_L$ is strongly non-self-averaging. We further show that the
probability distribution of $D_L$ has a log-normal left tail and a highly
singular, one-sided log-stable right tail reminiscent of a Lifshitz
singularity.
