Where Did the Brownian Particle Go?

Consider the radial projection onto the unit sphereof the path a $d$-dimensional Brownian motion $W$,started at the center of the sphere and run for unit time. Given the occupation measure $mu$ of this projectedpath, what can be said about the terminal point $W(1)$, or about therange of the original path? In any dimension, for each Borel set $A$ in $S^{d-1}$, the conditional probability that the projection of $W(1)$ is in $A$ given $mu(A)$ is just $mu(A)$. Nevertheless, in dimension $d ge 3$, both the range and the terminal point of $W$ can be recovered with probability 1 from $mu$. In particular, for $d ge 3$ the conditional law of the projection of $W(1)$ given $mu$ is not $mu$.In dimension 2 we conjecture thatthe projection of $W(1)$ cannot be recovered almost surely from $mu$,and show that the conditional law of the projection of $W(1)$ given $mu$ is not $mu$.