Some years ago, with A. J. McKane, I looked at a very simple problem of a single particle moving in a potential well and driven by a "coloured" external noise (i.e. a noise with a non-zero correlation time). We developed an "instanton" or "optimal path" method for determining the activation rate over a potential barrier. In contrast to "white" noise, it turns out that the activation rate depends on the whole shape of the potential. Consequently, the activation rate over a barrier can be different for the two crossing directions even when the barrier height is the same. It follows that coloured noise can drive a non-zero current in a system with a periodic but asymmetric potential, providing a type of "thermal ratchet", which is thought to be a possible mechanism for the transport of biological molecules.
My recent interests in stochastic processes centre on the newly discovered phenomenon of nontrivial "persistence" in coarsening processes. The simplest example might be the (Glauber) dynamics of the one-dimensional Ising model at T=0 , starting from a random initial configuration. Coarsening proceeds by annihilation of domain walls through the annihilation-diffusion process A + A -> 0 . One can ask the question, "What fraction of the spins has never flipped up to time t?". It turns out that in this model the fraction of "persistent spins" decreases like the minus 3/8 power of t. The generic behaviour, however, involves a nontrivial exponent theta which is not, in general, a rational fraction. A simple coarsening system is the ubiquitous diffusion equation, in which the stochastic element is associated entirely with the random initial conditions. In this system it is found that the fraction of space points where the field has always been above or below the mean decays as a power of time, the power being a nontrivial exponent theta (cond-mat/9605084). The "persistent set" comprised of these points has, furthermore, a fractal character, illustrated by some nice pictures.
These general ideas can be extended to any coarsening system. Interesting extensions include the total magnetization of a ferromagnet at its critical point (cond-mat/9606123 and 9702203), and the dynamics of fluctuating interfaces (cond-mat/9704238).
Related phenomena can be observed in the asymptotic behaviour of various reaction-diffusion processes.