Quenched distributions for the maximum, minimum and local time of the Brox diffusion

Abstract

After leaving fixed the environment, which is called the quenchend case, we give explicitly the distribution function of the maximum and the minimum of the Brox diffusion at first time it reaches a barrier. We also give explicit quenched formulæ for the distribution function of the local time of the Brox process at first hitting time of a constant, and at first exit time from an interval. To do that, we use the distribution functions of the maximum and of the minimum of the Brownian motion, as well as the local time of the Brownian motion. The main idea is to use the fact that the Brox diffusion can be written in terms of a time-change of a standard Brownian motion, and also to work with specific stopping times, namely, the first hitting time and exit time from an interval. As a bonus, we provide proofs of known formulas for the Brownian motion.