Planar Brownian motion

Planar Brownian motion is the random vector $(B_t^{(1)},B_t^{(2)})$ formed from two independent standard one-dimensional Brownian motions.

Applying any fixed orthogonal transformation to planar Brownian motion produces another planar Brownian motion. In particular, rotationally invariant initial data produce rotationally invariant hitting distributions.

Planar Brownian motion almost surely visits every neighborhood of every point. Equivalently, it hits every disc almost surely, although it does not hit any prescribed point with positive probability.

To couple Brownian motions starting at points exchanged by reflection in a hyperplane, reflect one path until the first path hits that hyperplane and then make the paths agree. Reflection invariance and the strong Markov property give the correct marginal laws.

The harmonic measure of a boundary set $A\subseteq \partial D$ viewed from $x\in D$ is the probability that Brownian motion started at $x$ first exits $D$ through $A$. As a function of $x$, it is harmonic in $D$.

For planar Brownian motion started at $x$ with $0<r<|x|<R$,This is optional stopping applied to the harmonic function $\log |x|$ on the annulus.