Microscopic models of random walks

We study various models of classical and quantum random walks from a microscopic point of view, i.e. trying to describe the random walkers as classical or quantum “open systems” interacting with “microscopic” reservoirs. We pay particular attention to models describing Brownian motion and diffusion in random inhomogeneous media and in media with random diffusivity, where in certain conditions anomalous diffusion might take place, non-ergodic behavior and aging might occur, as well as plenty of memory effects governs non-Markovian behavior. Such models find applications in biophysics, classical statistical physics, physics of activated carbon, nano-photonics, economy on the classical side, and in the physics of ultracold quantum gases on the other.