A density for the local time of the Brox diffusion

TL;DR

This paper derives an explicit probability density for the local time of the Brox diffusion at first passage times, enabling analysis of visitation patterns and environmental extrema.

Contribution

It provides a novel explicit density formula for the local time of Brox diffusion at first passage times, linking environmental features to visitation behavior.

Findings

Explicit density formula for local time at first passage times

Connection between environment extrema and visitation points

Method to compute moments of local time

Abstract

We give explictly the probability density of the local time of the Brox diffusion at first passage times. Such formula is used to find the moments and to related the minima and maxima of the environment to the most and least visted points of the diffusion.