Let $D \subset \mathbb{C}$ be a domain and $B_t$ a planar Brownian motion, and $p_D\left(t,z,w\right)$ be the transtion subdensity of it conditioned on that it does not leave $D$. Also the transition density of Brownian motion in this case is given by $p\left(t,z,w\right)=\frac{1}{2\pi t}exp\{-\frac{|z-w|^2}{2t}\}$.
Now define the Green's function of Brownian motion on $D$ as $$G\left(z,w\right)=\pi\int_0^\infty p_D\left(t,z,w\right)dt.$$
How do I show that $G\left(z,w\right)$ goes to $-log|z-w|$ as $w \to z$? For convenience let $z$ be fixed so that $G$ is a function of $w$.
I understand that the conclusion might be got with a purely analytic approach using differential equation theory. However, this time I wish to get a probabilistic explanation and link the log function with the expression given. I guess there is a relationship between $p_D\left(t,z,w\right)$ and $p\left(t,z,w\right)$, but I can't really figure out anything right now.
Thanks in advance.