Conditioned Brownian motion

Remarks

A positive harmonic function on $D$ is excessive for killed Brownian motion, so the h-transform is defined; the h-process may have a finite lifetime. Example: $D = (0, \infty)$ and $h(x) = x$. Here $h$ is positive and harmonic, and the h-process is the three-dimensional Bessel process: Brownian motion conditioned never to hit $0$. Example: for the unit disc and $h(z) = (1 - \abs{z}^2)/\abs{\zeta - z}^2$, the Poisson kernel with pole at $\zeta$ on the unit circle, the h-process is Brownian motion conditioned to exit the disc at $\zeta$. Non-example: $h(x) = x$ on $D = \R$ is harmonic but takes negative values, so the formula does not give a family of positive kernels. More generally, conditioning to exit at a point of the Martin boundary uses the minimal harmonic function with pole at that point.