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].
This work was partially supported by ONR Grant N00014-95-1-0224.
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Anderson, R.F., Molchanov, S.A. (1997). One Dimensional Random Walk in a Random Medium. In: Molchanov, S.A., Woyczynski, W.A. (eds) Stochastic Models in Geosystems. The IMA Volumes in Mathematics and its Applications, vol 85. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8500-4_2
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