@article{EJP83,
author = {Robin Pemantle and Yuval Peres and Jim Pitman and Marc Yor},
title = {Where Did the Brownian Particle Go?},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {6},
year = {2001},
keywords = {Brownian motion, conditional distribution of a path given its occupation measure, radial projection.},
abstract = {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 $A$ 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 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 $mu$.},
pages = {no. 10, 1-22},
issn = {1083-6489},
doi = {10.1214/EJP.v6-83},
url = {http://ejp.ejpecp.org/article/view/83}}