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 $V_{x,u} = x + W_{uτ(x)},$ $0\le u\le 1$, where $W_\bullet$ is the standard Brownian motion process, $x>0$ and $τ(x) =\inf\{t>0 : W_{t}=-x\}$.

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