On the distribution of the Brownian motion process on its way to hitting zero

We present functional versions of recent results on the univariate distributions of the process $V_{x,u}=x+W_{u\tau (x)},$ $0\le u\le 1$, where $W_{\bullet }$ is the standard Brownian motion process, $x>0$ and $\tau (x) =\inf{t>0 : W_{t}=-x}$.