  Consider the radial projection onto the unit sphere of 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 projected path, what can
be said about the terminal point W(1), or about the range of the original path?
In any dimension, for each Borel set A subseteq 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>=3, both the range and the terminal point of W can
be recovered with probability 1 from mu. In particular, for d>=3 the
conditional law of the projection of W(1) given mu is not mu. In dimension~2 we
conjecture that the 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.
