Distribution of the Brownian motion on its way to hitting zero

Abstract

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

7 pages, final version