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

An ${\mathbb{R}}^d$-valued process $(X_t{)}_{t\ge 0}$ is a Brownian motion started at $x$ when:

The paths of $U{X}_t$ are continuous, and linear transformation preserves independence of increments. Moreover,because $U$ is an orthogonal matrix. Thus all defining properties are preserved, proving the orthogonal invariance of Brownian motion:

Fix a closed ball $\overline{B}(x,r)\subset D$ and let $\tau$ be its first exit time. By the Strong Markov property, conditioning at $\tau$ givesThe orthogonal invariance of Brownian motion implies that $X_{\tau }$ is uniformly distributed on the sphere $\partial B(x,r)$. HenceThus $\varphi$ has the mean value property on every ball compactly contained in $D$. Since $0\le \varphi \le 1$, the mean-value characterization of harmonic functions yieldsThis function is the harmonic measure of $A$ viewed from $x$.

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,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.