Position of Brownian motion at exit time from the upper half plane

Question

I am currently reading some books on SLE and struggling on some problems regarding Brownian motion.

For a Brownian motion in $\mathbb{R}^2$ starting from $(x,y)$, I don't know how to find the distribution of $B_T$,

where $T:= \inf \{ t \geq 0 : B_t \not \in \mathbb{H}\}$, $\mathbb{H}:= \{ (x,y) \in \mathbb{R}^2 : y >0 \}$.

Any ideas on how to approach this problem?

Answer 1

If $D$ is a domain in $\mathbb{R}^n$, and $x_0 \in D$ is a point, then the distribution of Brownian motion started at $x_0$ stopped at the time when it leaves $D$ is the harmonic measure of $\partial D$ with base point $x_0$.

In the plane you can often use conformal invariance to calculate harmonic measure. Since you know that the harmonic measure of the unit circle with base point $0$ is just rescaled length measure, you can solve your problem by mapping the upper half plane to the unit disk, where $x+iy$ maps to $0$. The distribution you are looking for is then just the pull-back of rescaled length measure on the circle.