One Dimensional Random Walk in a Random Medium

Abstract

One of the fundamental problems in random medium (RM) theory is the relation between homogenization of the RM (i.e. the ability to describe such media at some scales as a homogeneous one) and localization. Homogenization describes transport processes in RM in terms of “effective” parameters, localization suppresses all forms of transport. For random walks in RM it is the relation between classical diffusive behavior of the random walk if time is large and the phenomenon of “trapping”, which can produce subdiffusion asymptotics for the random walk. In the one-dimensional case, which is of course the simplest one in RM theory, under some restrictions on the random transition probabilities the homogenization theorems was proved by S. Kozlov [10] as an example of a more general theory. A general and elementary introduction to homogenization theory can be found in S. Molchanov [2]. There are deep relations between random walks in one dimensional RM and scattering of waves (say, seismic waves) in random layered media. See [3].KeywordsRandom WalkSeismic WaveRandom EnvironmentRandom MediumStokes DriftThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.