Let $D$ be a domain in $\mathbb{R}^d$ and $A$ a measurable subset of its boundary $\partial D$. For $x \in D$ define $$\phi (x) = \mathbb{P}(X_T\in A) $$ where $(X_t)$ is a Brownian Motion in $\mathbb{R}^d$ starting from $x$ and $T=\inf\{t \geq 0 : X_t \not\in D\}$ is the exit time from D.
1) Show that $\phi$ is harmonic on D.
2) Find $\phi$ in the case $d=2$ for $D= \{(x,y):y>0\}, A=\{(x,0):x>0\}$
So far I have proved (1) by using the strong markov property and the fact that a finite function is harmonic on D if for every point $x\in D$ it is equal to its average on the sphere $S(x,r)$ for every radius $r>0$ such that $S(x,r) \subset D$ (even though I was not very formal so any help about this is appreciated too).
On the other hand I don't have any idea how to prove (2).
Any help appreciated, thanks!
If you think of the 2-dimensional Brownian motion as a pair $(B^1_t,B^2_t)$ of independent 1-dimensional Brownian motions, then $T$ is the first time that $B^2$ is equal to $0$. As such, $T$ has a known distribution, and is independent of $B^1$.