On birth and death processes in symmetric random environment
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 integers j and \((j+1)\) a rate \(\lambda_ j>0\) is attached. These \(\{\lambda_ j:\) j integral\(\}\) are taken as independent, identically distributed random variables, and represent the medium. For given values \(\lambda_ j\), X(t) is a Markov chain in continuous time which jumps from j to \((j+1)\) and from \((j+1)\) to j at the same rate \(\lambda_ j.\) We show that in many cases there exists normalizing constants \(\gamma\) (t) (which tend to \(\infty\) with t) such that the distribution of X(t)/\(\gamma\) (t), or more generally of the whole process \(\{X(st)/\gamma (t)\}_{s\geq 0}\), converges to a limit as \(t\to \infty\). The limit process is continuous and self-similar.
scientific article; zbMATH DE number 3942673
scientific article