The normalized Brownian excursion (|B|^{left[g_{1}, d_{1} ight]}) is a (mathrm{BES}^{3}) bridge from (x=0) to (y=0) with (t=1).

The normalized Brownian excursion \(|B|^{\left[g_{1}, d_{1}\right]}\) is a \(\mathrm{BES}^{3}\) bridge from \(x=0\) to \(y=0\) with \(t=1\).

\[\begin{equation*}
X_{t}^{[a, b]}=\frac{1}{\sqrt{b-a}} X_{a+t(b-a)}, 0 \leq t \leq 1 \tag{4.3.1}
\end{equation*}\]