Distribution of the Brownian motion on its way to hitting zero

Abstract

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