Transition density of Brownian motion in half space

Question

Let $H=\{x\in \mathcal R^d:x_d\geq0\}$ is the half space of $d$-dimensional Euclidean space, it is said that there is a explicit transition density formula for killed Brownian motion in $H$, but i can't find the formula. Any help please. Thanks very much!

Answer 1

One result that might help is mentioned in Continuous Martingales and Brownian Motion by Revuz & Yor. They note (e.g. page 87, 494) that $$ p_t(x,y) = \frac{1}{\sqrt{2\pi t}}\left[ \exp\left( \frac{-1}{2t}(y-x)^2 \right) - \exp\left( \frac{-1}{2t}(y+x)^2 \right) \right] $$ for $x,y>0$ is the transition density of Brownian motion killed when it reaches 0.