Killed Brownian transition density ($p_D(t,z,w)$)

The area density of ${\mathbb{P}}_z(B_t\in dw,t<T_D)$ for planar Brownian motion killed on its first exit from $D$. Its total mass is the survival probability, not generally one. Integrating this density over time gives the Green function of killed planar Brownian motion.