  Sinai's model of diffusion in one-dimension with random local bias is studied
by a real space renormalization group which yields exact results at long times.
The effects of an additional small uniform bias force are also studied. We
obtain analytically the scaling form of the distribution of the position $x(t)$
of a particle, the probability of it not returning to the origin and the
distributions of first passage times, in an infinite sample as well as in the
presence of a boundary and in a finite size sample. We compute the distribution
of meeting time of two particles. We also obtain a detailed analytic
description of thermally averaged trajectories: we compute the distributions of
the number of returns and of the number of jumps forward. They obey
multifractal scaling, characterized by generalized persistence exponents
$\theta(g)$ which we compute. With a small bias, the number of returns is
finite, characterized by a universal scaling function. The statistics of the
successive times of return of thermally averaged trajectories is obtained. The
two time distribution of the positions of a particle, $x(t)$ and $x(t')$
($t>t'$) is computed exactly. It exhibits ``aging'' with several regimes:
without a bias, for $t-t' \sim t'^\alpha, \alpha > 1$, it exhibits a $(\ln
t)/(\ln t')$ scaling, with a novel singularity at rescaled positions
$x(t)=x(t')$. For closer times $\alpha<1$ there is a quasi-equilibrium regime
with $\ln(t-t')/\ln t'$ scaling. The crossover to a $t/t'$ aging form under a
small bias is obtained analytically. Rare events, e.g. splitting of the thermal
packet between wells, are also studied. Connections with the Green's function
of a 1D Schr\"odinger problem and quantum spin chains are discussed.
