Abstract
We prove a limit theorem for a process in a random one-dimensional medium, which has been considered before as a model for hopping conduction in a disordered medium. To the edge between the two integersj and (j+ 1) a rate λj > 0 is attached. Theseλ j :j integral are taken as independent, identically distributed random variables, and represent the medium. For given values λj, X(t) is a Markov chain in continuous time which jumps fromj to (j + 1) and from (j + 1) toj at the same rate λj. We show that in many cases there exists normalizing constants y(t) (which tend to oo witht) such that the distribution of X(t)/γ(t), or more generally of the whole processX(st)/γ(t) S⩾0, converges to a limit as t→ ∞. The limit process is continuous and self-similar.
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Kawazu, K., Kesten, H. On birth and death processes in symmetric random environment. J Stat Phys 37, 561–576 (1984). https://doi.org/10.1007/BF01010495
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DOI: https://doi.org/10.1007/BF01010495