  Sinai's model of diffusion in one-dimension with random local bias is studied
by a real space renormalization group which yields asymptotically exact long
time results. The distribution of the position of a particle and the
probability of it not returning to the origin are obtained, as well as the
two-time distribution which exhibits "aging" with $\frac{\ln t}{\ln t'}$
scaling and a singularity at $\ln t =\ln t'$. The effects of a small uniform
force are also studied. Extension to motion of many domain walls yields
non-equilibrium time dependent correlations for the 1D random field Ising model
with Glauber dynamics and "persistence" exponents of 1D reaction-diffusion
models with random forces.
