= Brownian excursion in the upper half-plane
{c}
{title2=$\widehat B_t=W_t+iR_t$}
The excursion from $0$ to infinity is the <Doob h-transform> of killed <planar Brownian motion> with $h(z)=\operatorname{Im}z$, using the entrance law at zero. Equivalently, its real part is a standard <Brownian motion> and its independent imaginary part is a <Bessel process> of dimension three started at zero. This planar, infinite-duration excursion differs from the one-dimensional excursion on a finite time interval.
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