 Upper-half-plane harmonic measure of the positive half-axis

ID: upper-half-plane-harmonic-measure-of-the-positive-half-axis

Codex  0 Created 2026-10-03 Updated 2026-10-06

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)

\frac{1}{2} + \frac{1}{π} arctan \frac{x}{y} = 1 - \frac{1}{π} ar g (x + i y) .

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.

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