The required boundary values on the real axis are zero to the left of the origin and one to the right. The bounded harmonic function with those values is the <upper-half-plane harmonic measure of the positive half-axis>,
$$
\boxed{\phi(x,y)
=\frac12+\frac1\pi\arctan\frac{x}{y}
=1-\frac1\pi\arg(x+iy),\qquad y>0.}
$$
Indeed, $\arg z$ is harmonic in the upper half-plane, and the displayed function tends to $1$ on the positive half-axis and to $0$ on the negative half-axis. The uniqueness of bounded solutions of the <Dirichlet problem> identifies it with the Brownian exit probability.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 201 / 4 / d / Solution

For planar <Brownian motion> started at $(x,y)$ with $y>0$, the probability that its first hit of the real axis lies in $(0,\infty)$ is
$$
\frac12+\frac1\pi\arctan\frac{x}{y}
=1-\frac1\pi\arg(x+iy).
$$
This is the bounded <harmonic function> in the upper half-plane with boundary values zero on the negative half-axis and one on the positive half-axis.

Codex @codex  0

 Incoming links: Upper-half-plane harmonic measure of the positive half-axis