Supported by The European Science Foundation (ESF) through its project "Random Dynamics in Spatially Extended Systems"
Much is now known on models of random walks in a symmetric random environment. This leads to a diffusion-type behaviour with effectively a non-random diffusion matrix. The phenomena of effective self-averaging for random walks in a symmetric random environment is closely related to the homogenization problem for second order elliptic operators with random coefficients. Most results in this area were obtained in 1980's. However, the general asymmetric case of a random walk in a random environment cannot be treated by similar techniques, and remains largely open in more than one dimension. There has recently been considerable progress in particular concerning random walks in random environment with ballistic behaviour and large deviation principles for random walks in random environments. But many questions are still open. For instance, it is still not known whether for large enough perturbations one can get non-diffusive behaviour for high dimension. Also, almost nothing is known rigorously in the two-dimensional case.
Of considerable interest are also random walks in a random potentials. Here, if the potential is time independent, the asymptotic behaviour can be non-diffusive, and particles move to a trap where they stay for a long time. At the same time, the important case of a time-dependent potential is much less understood, although there are interesting recent developments.
Some closely related topics are: Random walks in random sceneries, directed polymers in random environments (which are connected with a wide class of growth models), and aging phenomena.