Brownian excursion in the upper half-plane (${\hat{B}}_t=W_t+i{R}_t$)

The excursion from $0$ to infinity is the Doob h-transform of killed planar Brownian motion with $h(z)=\mathrm{Im}z$, 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.

For a compact H-hull avoiding zero, let ${\psi }_A$ map its complement to the half-plane, fix zero, and have derivative one at infinity. The Brownian excursion in the upper half-plane avoids $A$ with probability ${\psi }_A^{\prime }(0)$. From an interior starting point $z$, the corresponding transformed process avoids $A$ with probability $\mathrm{Im}{g}_A(z)/\mathrm{Im}z$; taking $z=i\epsilon$ and $\epsilon \downarrow 0$ gives the boundary derivative.

For a compact H-hull $A$ whose closure avoids a real boundary point $x$, the Brownian excursion in the upper half-plane from $x$ to infinity avoids $A$ with the displayed probability. From an interior point $z$, the corresponding Doob h-transform avoidance probability is $\mathrm{Im}{g}_A(z)/\mathrm{Im}z$. Letting $z=x+i\epsilon$ approach the boundary and using the Schwarz reflection principle gives the derivative. In particular $0<g_A^{\prime }(x)\le 1$ at every regular real boundary point away from the hull.