It is known (Theorem 3.46 in Peres-Mörters) that the harmonic measure of a set $A$ from infinity is well-defined by taking the limit as $x\to \infty$ of a Brownian motion started at $x$, or by averaging the starting point on a sphere. This works in arbitrary dimension.
Can the path itself be defined, rather than just the hitting distribution?
Concretely, is it the same as running Brownian motion on the sphere (by stereographic projection) from the north pole until it hits $A$? Does it work then in all dimensions or just in dimension 2?
I think in dimension 2, defining the path via stereographic projection of the Brownian motion on the sphere does work. In higher dimensions, Brownian motion is not invariant under general conformal transformations. However, you can probably exploit the fact that Brownian motion in dimension $\ge 3$ is transient, and that it is symmetric with respect to time reversal. I.e., start at a random point and run two independent Brownian motions, then use one as the future, the other as the past of the Brownian motion started at $\infty$. (Incidentally, the transience of Brownian motion shows that $\lim_{t \to \infty} B_t = \infty$, which I think shows that this cannot be done via stereographic projection, since you would get a closed path starting and ending at the north pole.)