Distribution of the Brownian motion on its way to hitting zero

TL;DR

This paper derives the probability density function of the position of a one-dimensional Brownian motion at a fraction of its hitting time to zero, providing insights into its path distribution before absorption.

Contribution

It explicitly characterizes the distribution of the Brownian motion at a fractional time before hitting zero, a novel result in stochastic process analysis.

Findings

Derived the density of B_{uτ} for u in (0,1).

Provided a new understanding of the path distribution of Brownian motion.

Enhanced the theoretical framework for hitting time analysis.

Abstract

For the one-dimensional Brownian motion $B=(B_t)_{t\ge 0}$, started at $x>0$, and the first hitting time $\tau=\inf\{t\ge 0:B_t=0\}$, we find the probability density of $B_{u\tau}$ for a $u\in(0,1)$, i.e. of the Brownian motion on its way to hitting zero.