A local time curiosity in random environment
A random environment is a system of i.i.d. random variables. A random process is in a random environment if it depends on parameters which are at every time instant given as a realization of the corresponding random variable. The most elementary discrete time random process in random environment is Sinai's random walk. Its continuous time counterpart is the so-called Brox's diffusion process, and the relation is similar to that between the usual random walk and the Brownian motion. Therefore, naturally, both the processes have the same limit distributions. And a similar behaviour would be also expected for the maximum local times which are the quantities defined for a discrete time process \(\{S_k\}_{k\geq 0}\) as \(L^*(n) = \sup_{x\in Z} \sum ^n_{k=0} 1_{\{S_k=x\}}.\) But, surprisingly, the asymptotic behaviour of the discrete and continuous time processes differ substantially which is the main result of the paper.
Sinai's random walk in random environment
Brox's diffusion with Brownian potential
scientific article; zbMATH DE number 1364495
scientific article