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Sinai's model of diffusion in one-dimension with random local bias is studiedby a real space renormalization group which yields asymptotically exact longtime results. The distribution of the position of a particle and theprobability of it not returning to the origin are obtained, as well as thetwo-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 uniformforce are also studied. Extension to motion of many domain walls yieldsnon-equilibrium time dependent correlations for the 1D random field Ising modelwith Glauber dynamics and "persistence" exponents of 1D reaction-diffusionmodels with random forces.Comment: 5 pages, 1 figures, RevTe

Random Walks, Reaction-Diffusion, and Nonequilibrium Dynamics of Spin Chains in One-Dimensional Random Environments