A local time curiosity in random environment

In random environments, the most elementary processes are Sinai’s simple random walk and Brox’s diffusion process, respectively in discrete and continuous time settings. The two processes are often considered as a kind of companions, somewhat in the same way as the usual random walk and Brownian motion are. In this paper, we study the maximum local times for the Sinai and Brox processes. A somewhat peculiar asymptotic behaviour is observed.